poimirlem27

Description: Lemma for poimir showing that the difference between admissible faces in the whole cube and admissible faces on the back face is even. Equation (7) of Kulpa p. 548. (Contributed by Brendan Leahy, 21-Aug-2020)

	Ref	Expression
Hypotheses	poimir.0	$⊢ φ \to N \in ℕ$
	poimirlem28.1	$⊢ p = 1^{st} (s) +_{f} ((2^{nd} (s) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (s) [(j + 1 \dots N)] \times \{0\})) \to B = C$
	poimirlem28.2	$⊢ (φ \land p : (1 \dots N) ⟶ (0 \dots K)) \to B \in (0 \dots N)$
	poimirlem28.3	$⊢ (φ \land (n \in (1 \dots N) \land p : (1 \dots N) ⟶ (0 \dots K) \land p (n) = 0)) \to B < n$
	poimirlem28.4	$⊢ (φ \land (n \in (1 \dots N) \land p : (1 \dots N) ⟶ (0 \dots K) \land p (n) = K)) \to B \neq n - 1$
Assertion	poimirlem27	$⊢ φ \to 2 ∥ (\|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}\|)$

Proof

Step	Hyp	Ref	Expression
1		poimir.0	$⊢ φ \to N \in ℕ$
2		poimirlem28.1	$⊢ p = 1^{st} (s) +_{f} ((2^{nd} (s) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (s) [(j + 1 \dots N)] \times \{0\})) \to B = C$
3		poimirlem28.2	$⊢ (φ \land p : (1 \dots N) ⟶ (0 \dots K)) \to B \in (0 \dots N)$
4		poimirlem28.3	$⊢ (φ \land (n \in (1 \dots N) \land p : (1 \dots N) ⟶ (0 \dots K) \land p (n) = 0)) \to B < n$
5		poimirlem28.4	$⊢ (φ \land (n \in (1 \dots N) \land p : (1 \dots N) ⟶ (0 \dots K) \land p (n) = K)) \to B \neq n - 1$
6		fzfi	$⊢ (0 \dots K) \in Fin$
7		fzfi	$⊢ (1 \dots N) \in Fin$
8		mapfi	$⊢ ((0 \dots K) \in Fin \land (1 \dots N) \in Fin) \to {(0 \dots K)}^{(1 \dots N)} \in Fin$
9	6 7 8	mp2an	$⊢ {(0 \dots K)}^{(1 \dots N)} \in Fin$
10		fzfi	$⊢ (0 \dots N - 1) \in Fin$
11		mapfi	$⊢ ({(0 \dots K)}^{(1 \dots N)} \in Fin \land (0 \dots N - 1) \in Fin) \to {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)} \in Fin$
12	9 10 11	mp2an	$⊢ {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)} \in Fin$
13	12	a1i	$⊢ φ \to {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)} \in Fin$
14		2z	$⊢ 2 \in ℤ$
15	14	a1i	$⊢ φ \to 2 \in ℤ$
16		fzofi	$⊢ (0 ..^K) \in Fin$
17		mapfi	$⊢ ((0 ..^K) \in Fin \land (1 \dots N) \in Fin) \to {(0 ..^K)}^{(1 \dots N)} \in Fin$
18	16 7 17	mp2an	$⊢ {(0 ..^K)}^{(1 \dots N)} \in Fin$
19		mapfi	$⊢ ((1 \dots N) \in Fin \land (1 \dots N) \in Fin) \to {(1 \dots N)}^{(1 \dots N)} \in Fin$
20	7 7 19	mp2an	$⊢ {(1 \dots N)}^{(1 \dots N)} \in Fin$
21		f1of	$⊢ f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N) \to f : (1 \dots N) ⟶ (1 \dots N)$
22	21	ss2abi	$⊢ \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \subseteq \{f \| f : (1 \dots N) ⟶ (1 \dots N)\}$
23		ovex	$⊢ (1 \dots N) \in V$
24	23 23	mapval	$⊢ {(1 \dots N)}^{(1 \dots N)} = \{f \| f : (1 \dots N) ⟶ (1 \dots N)\}$
25	22 24	sseqtrri	$⊢ \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \subseteq {(1 \dots N)}^{(1 \dots N)}$
26		ssfi	$⊢ ({(1 \dots N)}^{(1 \dots N)} \in Fin \land \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \subseteq {(1 \dots N)}^{(1 \dots N)}) \to \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
27	20 25 26	mp2an	$⊢ \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
28		xpfi	$⊢ ({(0 ..^K)}^{(1 \dots N)} \in Fin \land \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin) \to ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
29	18 27 28	mp2an	$⊢ ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
30		fzfi	$⊢ (0 \dots N) \in Fin$
31		xpfi	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin \land (0 \dots N) \in Fin) \to (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin$
32	29 30 31	mp2an	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin$
33		rabfi	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} \in Fin$
34	32 33	ax-mp	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} \in Fin$
35		hashcl	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} \in Fin \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\| \in ℕ_{0}$
36	35	nn0zd	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} \in Fin \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\| \in ℤ$
37	34 36	mp1i	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\| \in ℤ$
38		dfrex2	$⊢ \exists t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \leftrightarrow \neg \forall t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \neg ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))$
39		nfv	$⊢ Ⅎ t (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}))$
40		nfcv	$⊢ \underline{Ⅎ} t 2$
41		nfcv	$⊢ \underline{Ⅎ} t ∥$
42		nfcv	$⊢ \underline{Ⅎ} t \|.\|$
43		nfrab1	$⊢ \underline{Ⅎ} t \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}$
44	42 43	nffv	$⊢ \underline{Ⅎ} t \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|$
45	40 41 44	nfbr	$⊢ Ⅎ t 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|$
46		neq0	$⊢ \neg \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} = \emptyset \leftrightarrow \exists s s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}$
47		iddvds	$⊢ 2 \in ℤ \to 2 ∥ 2$
48	14 47	ax-mp	$⊢ 2 ∥ 2$
49		vex	$⊢ s \in V$
50		hashsng	$⊢ s \in V \to \|\{s\}\| = 1$
51	49 50	ax-mp	$⊢ \|\{s\}\| = 1$
52	51	oveq2i	$⊢ 1 + \|\{s\}\| = 1 + 1$
53		df-2	$⊢ 2 = 1 + 1$
54	52 53	eqtr4i	$⊢ 1 + \|\{s\}\| = 2$
55	48 54	breqtrri	$⊢ 2 ∥ (1 + \|\{s\}\|)$
56		rabfi	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \in Fin$
57		diffi	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \in Fin \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\} \in Fin$
58	32 56 57	mp2b	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\} \in Fin$
59		snfi	$⊢ \{s\} \in Fin$
60		disjdifr	$⊢ (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}) \cap \{s\} = \emptyset$
61		hashun	$⊢ (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\} \in Fin \land \{s\} \in Fin \land (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}) \cap \{s\} = \emptyset) \to \|(\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}) \cup \{s\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| + \|\{s\}\|$
62	58 59 60 61	mp3an	$⊢ \|(\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}) \cup \{s\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| + \|\{s\}\|$
63		difsnid	$⊢ s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \to (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}) \cup \{s\} = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}$
64	63	fveq2d	$⊢ s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \to \|(\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}) \cup \{s\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|$
65	62 64	eqtr3id	$⊢ s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| + \|\{s\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|$
66	65	adantl	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| + \|\{s\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|$
67	1	ad3antrrr	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to N \in ℕ$
68		fveq2	$⊢ t = u \to 2^{nd} (t) = 2^{nd} (u)$
69	68	breq2d	$⊢ t = u \to (y < 2^{nd} (t) \leftrightarrow y < 2^{nd} (u))$
70	69	ifbid	$⊢ t = u \to if (y < 2^{nd} (t), y, y + 1) = if (y < 2^{nd} (u), y, y + 1)$
71	70	csbeq1d	$⊢ t = u \to ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (y < 2^{nd} (u), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))$
72		2fveq3	$⊢ t = u \to 1^{st} (1^{st} (t)) = 1^{st} (1^{st} (u))$
73		2fveq3	$⊢ t = u \to 2^{nd} (1^{st} (t)) = 2^{nd} (1^{st} (u))$
74	73	imaeq1d	$⊢ t = u \to 2^{nd} (1^{st} (t)) [(1 \dots j)] = 2^{nd} (1^{st} (u)) [(1 \dots j)]$
75	74	xpeq1d	$⊢ t = u \to 2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\} = 2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}$
76	73	imaeq1d	$⊢ t = u \to 2^{nd} (1^{st} (t)) [(j + 1 \dots N)] = 2^{nd} (1^{st} (u)) [(j + 1 \dots N)]$
77	76	xpeq1d	$⊢ t = u \to 2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\} = 2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\}$
78	75 77	uneq12d	$⊢ t = u \to (2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}) = (2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\})$
79	72 78	oveq12d	$⊢ t = u \to 1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})) = 1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\}))$
80	79	csbeq2dv	$⊢ t = u \to ⦋ if (y < 2^{nd} (u), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (y < 2^{nd} (u), y, y + 1) / j ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\})))$
81	71 80	eqtrd	$⊢ t = u \to ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (y < 2^{nd} (u), y, y + 1) / j ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\})))$
82	81	mpteq2dv	$⊢ t = u \to (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (u), y, y + 1) / j ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\}))))$
83		breq1	$⊢ y = w \to (y < 2^{nd} (u) \leftrightarrow w < 2^{nd} (u))$
84		id	$⊢ y = w \to y = w$
85		oveq1	$⊢ y = w \to y + 1 = w + 1$
86	83 84 85	ifbieq12d	$⊢ y = w \to if (y < 2^{nd} (u), y, y + 1) = if (w < 2^{nd} (u), w, w + 1)$
87	86	csbeq1d	$⊢ y = w \to ⦋ if (y < 2^{nd} (u), y, y + 1) / j ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (w < 2^{nd} (u), w, w + 1) / j ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\})))$
88		oveq2	$⊢ j = i \to (1 \dots j) = (1 \dots i)$
89	88	imaeq2d	$⊢ j = i \to 2^{nd} (1^{st} (u)) [(1 \dots j)] = 2^{nd} (1^{st} (u)) [(1 \dots i)]$
90	89	xpeq1d	$⊢ j = i \to 2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\} = 2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}$
91		oveq1	$⊢ j = i \to j + 1 = i + 1$
92	91	oveq1d	$⊢ j = i \to (j + 1 \dots N) = (i + 1 \dots N)$
93	92	imaeq2d	$⊢ j = i \to 2^{nd} (1^{st} (u)) [(j + 1 \dots N)] = 2^{nd} (1^{st} (u)) [(i + 1 \dots N)]$
94	93	xpeq1d	$⊢ j = i \to 2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\} = 2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\}$
95	90 94	uneq12d	$⊢ j = i \to (2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\}) = (2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\})$
96	95	oveq2d	$⊢ j = i \to 1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\})) = 1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\}))$
97	96	cbvcsbv	$⊢ ⦋ if (w < 2^{nd} (u), w, w + 1) / j ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (w < 2^{nd} (u), w, w + 1) / i ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\})))$
98	87 97	eqtrdi	$⊢ y = w \to ⦋ if (y < 2^{nd} (u), y, y + 1) / j ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (w < 2^{nd} (u), w, w + 1) / i ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\})))$
99	98	cbvmptv	$⊢ (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (u), y, y + 1) / j ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(j + 1 \dots N)] \times \{0\})))) = (w \in (0 \dots N - 1) ⟼ ⦋ if (w < 2^{nd} (u), w, w + 1) / i ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\}))))$
100	82 99	eqtrdi	$⊢ t = u \to (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) = (w \in (0 \dots N - 1) ⟼ ⦋ if (w < 2^{nd} (u), w, w + 1) / i ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\}))))$
101	100	eqeq2d	$⊢ t = u \to (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \leftrightarrow x = (w \in (0 \dots N - 1) ⟼ ⦋ if (w < 2^{nd} (u), w, w + 1) / i ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\})))))$
102	101	cbvrabv	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} = \{u \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (w \in (0 \dots N - 1) ⟼ ⦋ if (w < 2^{nd} (u), w, w + 1) / i ⦌ (1^{st} (1^{st} (u)) +_{f} ((2^{nd} (1^{st} (u)) [(1 \dots i)] \times \{1\}) \cup (2^{nd} (1^{st} (u)) [(i + 1 \dots N)] \times \{0\}))))\}$
103		elmapi	$⊢ x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) \to x : (0 \dots N - 1) ⟶ {(0 \dots K)}^{(1 \dots N)}$
104	103	ad3antlr	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to x : (0 \dots N - 1) ⟶ {(0 \dots K)}^{(1 \dots N)}$
105		simpr	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}$
106		simpl	$⊢ (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K) \to \exists p \in ran x p (n) \neq 0$
107	106	ralimi	$⊢ \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K) \to \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0$
108	107	ad2antlr	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0$
109		fveq2	$⊢ n = m \to p (n) = p (m)$
110	109	neeq1d	$⊢ n = m \to (p (n) \neq 0 \leftrightarrow p (m) \neq 0)$
111	110	rexbidv	$⊢ n = m \to (\exists p \in ran x p (n) \neq 0 \leftrightarrow \exists p \in ran x p (m) \neq 0)$
112		fveq1	$⊢ p = q \to p (m) = q (m)$
113	112	neeq1d	$⊢ p = q \to (p (m) \neq 0 \leftrightarrow q (m) \neq 0)$
114	113	cbvrexvw	$⊢ \exists p \in ran x p (m) \neq 0 \leftrightarrow \exists q \in ran x q (m) \neq 0$
115	111 114	bitrdi	$⊢ n = m \to (\exists p \in ran x p (n) \neq 0 \leftrightarrow \exists q \in ran x q (m) \neq 0)$
116	115	rspccva	$⊢ (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \land m \in (1 \dots N)) \to \exists q \in ran x q (m) \neq 0$
117	108 116	sylan	$⊢ ((((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \land m \in (1 \dots N)) \to \exists q \in ran x q (m) \neq 0$
118		simpr	$⊢ (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K) \to \exists p \in ran x p (n) \neq K$
119	118	ralimi	$⊢ \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K) \to \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq K$
120	119	ad2antlr	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq K$
121	109	neeq1d	$⊢ n = m \to (p (n) \neq K \leftrightarrow p (m) \neq K)$
122	121	rexbidv	$⊢ n = m \to (\exists p \in ran x p (n) \neq K \leftrightarrow \exists p \in ran x p (m) \neq K)$
123	112	neeq1d	$⊢ p = q \to (p (m) \neq K \leftrightarrow q (m) \neq K)$
124	123	cbvrexvw	$⊢ \exists p \in ran x p (m) \neq K \leftrightarrow \exists q \in ran x q (m) \neq K$
125	122 124	bitrdi	$⊢ n = m \to (\exists p \in ran x p (n) \neq K \leftrightarrow \exists q \in ran x q (m) \neq K)$
126	125	rspccva	$⊢ (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq K \land m \in (1 \dots N)) \to \exists q \in ran x q (m) \neq K$
127	120 126	sylan	$⊢ ((((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \land m \in (1 \dots N)) \to \exists q \in ran x q (m) \neq K$
128	67 102 104 105 117 127	poimirlem22	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to \exists! z \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} z \neq s$
129		eldifsn	$⊢ z \in (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}) \leftrightarrow (z \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \land z \neq s)$
130	129	eubii	$⊢ \exists! z z \in (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}) \leftrightarrow \exists! z (z \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \land z \neq s)$
131	58	elexi	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\} \in V$
132		euhash1	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\} \in V \to (\|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| = 1 \leftrightarrow \exists! z z \in (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}))$
133	131 132	ax-mp	$⊢ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| = 1 \leftrightarrow \exists! z z \in (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\})$
134		df-reu	$⊢ \exists! z \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} z \neq s \leftrightarrow \exists! z (z \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \land z \neq s)$
135	130 133 134	3bitr4ri	$⊢ \exists! z \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} z \neq s \leftrightarrow \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| = 1$
136	128 135	sylib	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| = 1$
137	136	oveq1d	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} ∖ \{s\}\| + \|\{s\}\| = 1 + \|\{s\}\|$
138	66 137	eqtr3d	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\| = 1 + \|\{s\}\|$
139	55 138	breqtrrid	$⊢ (((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \land s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|$
140	139	ex	$⊢ ((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to (s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|)$
141	140	exlimdv	$⊢ ((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to (\exists s s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|)$
142	46 141	biimtrid	$⊢ ((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to (\neg \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} = \emptyset \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|)$
143		dvds0	$⊢ 2 \in ℤ \to 2 ∥ 0$
144	14 143	ax-mp	$⊢ 2 ∥ 0$
145		hash0	$⊢ \|\emptyset\| = 0$
146	144 145	breqtrri	$⊢ 2 ∥ \|\emptyset\|$
147		fveq2	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} = \emptyset \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\| = \|\emptyset\|$
148	146 147	breqtrrid	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} = \emptyset \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|$
149	142 148	pm2.61d2	$⊢ ((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|$
150	149	ex	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to (\forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|)$
151	150	adantld	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to (((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\|)$
152		iba	$⊢ ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \leftrightarrow (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))))$
153	152	rabbidv	$⊢ ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}$
154	153	fveq2d	$⊢ ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|$
155	154	breq2d	$⊢ ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to (2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}\| \leftrightarrow 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|)$
156	151 155	mpbidi	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to (((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|)$
157	156	a1d	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to (t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to (((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|))$
158	39 45 157	rexlimd	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to (\exists t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|)$
159	38 158	biimtrrid	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to (\neg \forall t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \neg ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|)$
160		simpr	$⊢ (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))) \to ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))$
161	160	con3i	$⊢ \neg ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to \neg (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))$
162	161	ralimi	$⊢ \forall t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \neg ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to \forall t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \neg (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))$
163		rabeq0	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} = \emptyset \leftrightarrow \forall t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \neg (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))$
164	162 163	sylibr	$⊢ \forall t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \neg ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} = \emptyset$
165	164	fveq2d	$⊢ \forall t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \neg ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\| = \|\emptyset\|$
166	146 165	breqtrrid	$⊢ \forall t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \neg ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|$
167	159 166	pm2.61d2	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to 2 ∥ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|$
168	13 15 37 167	fsumdvds	$⊢ φ \to 2 ∥ \sum_{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}} \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|$
169		rabfi	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} \in Fin$
170	32 169	ax-mp	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} \in Fin$
171		simp1	$⊢ (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N) \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C$
172		sneq	$⊢ 2^{nd} (t) = N \to \{2^{nd} (t)\} = \{N\}$
173	172	difeq2d	$⊢ 2^{nd} (t) = N \to (0 \dots N) ∖ \{2^{nd} (t)\} = (0 \dots N) ∖ \{N\}$
174		difun2	$⊢ ((0 \dots N - 1) \cup \{N\}) ∖ \{N\} = (0 \dots N - 1) ∖ \{N\}$
175	1	nnnn0d	$⊢ φ \to N \in ℕ_{0}$
176		nn0uz	$⊢ ℕ_{0} = ℤ_{\geq 0}$
177	175 176	eleqtrdi	$⊢ φ \to N \in ℤ_{\geq 0}$
178		fzm1	$⊢ N \in ℤ_{\geq 0} \to (n \in (0 \dots N) \leftrightarrow (n \in (0 \dots N - 1) \lor n = N))$
179	177 178	syl	$⊢ φ \to (n \in (0 \dots N) \leftrightarrow (n \in (0 \dots N - 1) \lor n = N))$
180		elun	$⊢ n \in ((0 \dots N - 1) \cup \{N\}) \leftrightarrow (n \in (0 \dots N - 1) \lor n \in \{N\})$
181		velsn	$⊢ n \in \{N\} \leftrightarrow n = N$
182	181	orbi2i	$⊢ (n \in (0 \dots N - 1) \lor n \in \{N\}) \leftrightarrow (n \in (0 \dots N - 1) \lor n = N)$
183	180 182	bitri	$⊢ n \in ((0 \dots N - 1) \cup \{N\}) \leftrightarrow (n \in (0 \dots N - 1) \lor n = N)$
184	179 183	bitr4di	$⊢ φ \to (n \in (0 \dots N) \leftrightarrow n \in ((0 \dots N - 1) \cup \{N\}))$
185	184	eqrdv	$⊢ φ \to (0 \dots N) = (0 \dots N - 1) \cup \{N\}$
186	185	difeq1d	$⊢ φ \to (0 \dots N) ∖ \{N\} = ((0 \dots N - 1) \cup \{N\}) ∖ \{N\}$
187	1	nnzd	$⊢ φ \to N \in ℤ$
188		uzid	$⊢ N \in ℤ \to N \in ℤ_{\geq N}$
189		uznfz	$⊢ N \in ℤ_{\geq N} \to \neg N \in (0 \dots N - 1)$
190	187 188 189	3syl	$⊢ φ \to \neg N \in (0 \dots N - 1)$
191		disjsn	$⊢ (0 \dots N - 1) \cap \{N\} = \emptyset \leftrightarrow \neg N \in (0 \dots N - 1)$
192		disj3	$⊢ (0 \dots N - 1) \cap \{N\} = \emptyset \leftrightarrow (0 \dots N - 1) = (0 \dots N - 1) ∖ \{N\}$
193	191 192	bitr3i	$⊢ \neg N \in (0 \dots N - 1) \leftrightarrow (0 \dots N - 1) = (0 \dots N - 1) ∖ \{N\}$
194	190 193	sylib	$⊢ φ \to (0 \dots N - 1) = (0 \dots N - 1) ∖ \{N\}$
195	174 186 194	3eqtr4a	$⊢ φ \to (0 \dots N) ∖ \{N\} = (0 \dots N - 1)$
196	173 195	sylan9eqr	$⊢ (φ \land 2^{nd} (t) = N) \to (0 \dots N) ∖ \{2^{nd} (t)\} = (0 \dots N - 1)$
197	196	rexeqdv	$⊢ (φ \land 2^{nd} (t) = N) \to (\exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C)$
198	197	biimprd	$⊢ (φ \land 2^{nd} (t) = N) \to (\exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \to \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
199	198	ralimdv	$⊢ (φ \land 2^{nd} (t) = N) \to (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
200	199	expimpd	$⊢ φ \to ((2^{nd} (t) = N \land \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C) \to \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
201	171 200	sylan2i	$⊢ φ \to ((2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \to \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
202	201	adantr	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \to \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
203	202	ss2rabdv	$⊢ φ \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \subseteq \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}$
204		hashssdif	$⊢ (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} \in Fin \land \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \subseteq \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\|$
205	170 203 204	sylancr	$⊢ φ \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\|$
206	1	adantr	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to N \in ℕ$
207	3	adantlr	$⊢ ((φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \land p : (1 \dots N) ⟶ (0 \dots K)) \to B \in (0 \dots N)$
208		xp1st	$⊢ t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})$
209		xp1st	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to 1^{st} (1^{st} (t)) \in ({(0 ..^K)}^{(1 \dots N)})$
210		elmapi	$⊢ 1^{st} (1^{st} (t)) \in ({(0 ..^K)}^{(1 \dots N)}) \to 1^{st} (1^{st} (t)) : (1 \dots N) ⟶ (0 ..^K)$
211	208 209 210	3syl	$⊢ t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 1^{st} (1^{st} (t)) : (1 \dots N) ⟶ (0 ..^K)$
212	211	adantl	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to 1^{st} (1^{st} (t)) : (1 \dots N) ⟶ (0 ..^K)$
213		xp2nd	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to 2^{nd} (1^{st} (t)) \in \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}$
214		fvex	$⊢ 2^{nd} (1^{st} (t)) \in V$
215		f1oeq1	$⊢ f = 2^{nd} (1^{st} (t)) \to (f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N) \leftrightarrow 2^{nd} (1^{st} (t)) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N))$
216	214 215	elab	$⊢ 2^{nd} (1^{st} (t)) \in \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \leftrightarrow 2^{nd} (1^{st} (t)) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)$
217	213 216	sylib	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to 2^{nd} (1^{st} (t)) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)$
218	208 217	syl	$⊢ t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 2^{nd} (1^{st} (t)) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)$
219	218	adantl	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to 2^{nd} (1^{st} (t)) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)$
220		xp2nd	$⊢ t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 2^{nd} (t) \in (0 \dots N)$
221	220	adantl	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to 2^{nd} (t) \in (0 \dots N)$
222	206 2 207 212 219 221	poimirlem24	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to (\exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \exists p \in ran x p (N) \neq 0)) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ ⟨1^{st} (1^{st} (t)), 2^{nd} (1^{st} (t))⟩ / s ⦌ C \land \neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))))$
223	208	adantl	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})$
224		1st2nd2	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to 1^{st} (t) = ⟨1^{st} (1^{st} (t)), 2^{nd} (1^{st} (t))⟩$
225	224	csbeq1d	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ ⟨1^{st} (1^{st} (t)), 2^{nd} (1^{st} (t))⟩ / s ⦌ C$
226	225	eqeq2d	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow i = ⦋ ⟨1^{st} (1^{st} (t)), 2^{nd} (1^{st} (t))⟩ / s ⦌ C)$
227	226	rexbidv	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to (\exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ ⟨1^{st} (1^{st} (t)), 2^{nd} (1^{st} (t))⟩ / s ⦌ C)$
228	227	ralbidv	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ ⟨1^{st} (1^{st} (t)), 2^{nd} (1^{st} (t))⟩ / s ⦌ C)$
229	228	anbi1d	$⊢ 1^{st} (t) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ ⟨1^{st} (1^{st} (t)), 2^{nd} (1^{st} (t))⟩ / s ⦌ C \land \neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))))$
230	223 229	syl	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ ⟨1^{st} (1^{st} (t)), 2^{nd} (1^{st} (t))⟩ / s ⦌ C \land \neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))))$
231	222 230	bitr4d	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to (\exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \exists p \in ran x p (N) \neq 0)) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))))$
232	103	frnd	$⊢ x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) \to ran x \subseteq {(0 \dots K)}^{(1 \dots N)}$
233	232	anim2i	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to (φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)})$
234		dfss3	$⊢ (0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \leftrightarrow \forall n \in (0 \dots N - 1) n \in ran (p \in ran x ⟼ B)$
235		vex	$⊢ n \in V$
236		eqid	$⊢ (p \in ran x ⟼ B) = (p \in ran x ⟼ B)$
237	236	elrnmpt	$⊢ n \in V \to (n \in ran (p \in ran x ⟼ B) \leftrightarrow \exists p \in ran x n = B)$
238	235 237	ax-mp	$⊢ n \in ran (p \in ran x ⟼ B) \leftrightarrow \exists p \in ran x n = B$
239	238	ralbii	$⊢ \forall n \in (0 \dots N - 1) n \in ran (p \in ran x ⟼ B) \leftrightarrow \forall n \in (0 \dots N - 1) \exists p \in ran x n = B$
240	234 239	sylbb	$⊢ (0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \to \forall n \in (0 \dots N - 1) \exists p \in ran x n = B$
241		1eluzge0	$⊢ 1 \in ℤ_{\geq 0}$
242		fzss1	$⊢ 1 \in ℤ_{\geq 0} \to (1 \dots N - 1) \subseteq (0 \dots N - 1)$
243		ssralv	$⊢ (1 \dots N - 1) \subseteq (0 \dots N - 1) \to (\forall n \in (0 \dots N - 1) \exists p \in ran x n = B \to \forall n \in (1 \dots N - 1) \exists p \in ran x n = B)$
244	241 242 243	mp2b	$⊢ \forall n \in (0 \dots N - 1) \exists p \in ran x n = B \to \forall n \in (1 \dots N - 1) \exists p \in ran x n = B$
245	1	nncnd	$⊢ φ \to N \in ℂ$
246		npcan1	$⊢ N \in ℂ \to N - 1 + 1 = N$
247	245 246	syl	$⊢ φ \to N - 1 + 1 = N$
248		peano2zm	$⊢ N \in ℤ \to N - 1 \in ℤ$
249		uzid	$⊢ N - 1 \in ℤ \to N - 1 \in ℤ_{\geq (N - 1)}$
250		peano2uz	$⊢ N - 1 \in ℤ_{\geq (N - 1)} \to N - 1 + 1 \in ℤ_{\geq (N - 1)}$
251	187 248 249 250	4syl	$⊢ φ \to N - 1 + 1 \in ℤ_{\geq (N - 1)}$
252	247 251	eqeltrrd	$⊢ φ \to N \in ℤ_{\geq (N - 1)}$
253		fzss2	$⊢ N \in ℤ_{\geq (N - 1)} \to (1 \dots N - 1) \subseteq (1 \dots N)$
254	252 253	syl	$⊢ φ \to (1 \dots N - 1) \subseteq (1 \dots N)$
255	254	sselda	$⊢ (φ \land n \in (1 \dots N - 1)) \to n \in (1 \dots N)$
256	255	adantlr	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N - 1)) \to n \in (1 \dots N)$
257		simplr	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N)) \to ran x \subseteq {(0 \dots K)}^{(1 \dots N)}$
258		ssel2	$⊢ (ran x \subseteq {(0 \dots K)}^{(1 \dots N)} \land p \in ran x) \to p \in ({(0 \dots K)}^{(1 \dots N)})$
259		elmapi	$⊢ p \in ({(0 \dots K)}^{(1 \dots N)}) \to p : (1 \dots N) ⟶ (0 \dots K)$
260	258 259	syl	$⊢ (ran x \subseteq {(0 \dots K)}^{(1 \dots N)} \land p \in ran x) \to p : (1 \dots N) ⟶ (0 \dots K)$
261	257 260	sylan	$⊢ (((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N)) \land p \in ran x) \to p : (1 \dots N) ⟶ (0 \dots K)$
262		elfzelz	$⊢ n \in (1 \dots N) \to n \in ℤ$
263	262	zred	$⊢ n \in (1 \dots N) \to n \in ℝ$
264	263	ltnrd	$⊢ n \in (1 \dots N) \to \neg n < n$
265		breq1	$⊢ n = B \to (n < n \leftrightarrow B < n)$
266	265	notbid	$⊢ n = B \to (\neg n < n \leftrightarrow \neg B < n)$
267	264 266	syl5ibcom	$⊢ n \in (1 \dots N) \to (n = B \to \neg B < n)$
268	267	necon2ad	$⊢ n \in (1 \dots N) \to (B < n \to n \neq B)$
269	268	3ad2ant1	$⊢ (n \in (1 \dots N) \land p : (1 \dots N) ⟶ (0 \dots K) \land p (n) = 0) \to (B < n \to n \neq B)$
270	269	adantl	$⊢ (φ \land (n \in (1 \dots N) \land p : (1 \dots N) ⟶ (0 \dots K) \land p (n) = 0)) \to (B < n \to n \neq B)$
271	4 270	mpd	$⊢ (φ \land (n \in (1 \dots N) \land p : (1 \dots N) ⟶ (0 \dots K) \land p (n) = 0)) \to n \neq B$
272	271	3exp2	$⊢ φ \to (n \in (1 \dots N) \to (p : (1 \dots N) ⟶ (0 \dots K) \to (p (n) = 0 \to n \neq B)))$
273	272	imp31	$⊢ ((φ \land n \in (1 \dots N)) \land p : (1 \dots N) ⟶ (0 \dots K)) \to (p (n) = 0 \to n \neq B)$
274	273	necon2d	$⊢ ((φ \land n \in (1 \dots N)) \land p : (1 \dots N) ⟶ (0 \dots K)) \to (n = B \to p (n) \neq 0)$
275	274	adantllr	$⊢ (((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N)) \land p : (1 \dots N) ⟶ (0 \dots K)) \to (n = B \to p (n) \neq 0)$
276	261 275	syldan	$⊢ (((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N)) \land p \in ran x) \to (n = B \to p (n) \neq 0)$
277	276	reximdva	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N)) \to (\exists p \in ran x n = B \to \exists p \in ran x p (n) \neq 0)$
278	256 277	syldan	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N - 1)) \to (\exists p \in ran x n = B \to \exists p \in ran x p (n) \neq 0)$
279	278	ralimdva	$⊢ (φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \to (\forall n \in (1 \dots N - 1) \exists p \in ran x n = B \to \forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0)$
280	279	imp	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (1 \dots N - 1) \exists p \in ran x n = B) \to \forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0$
281	244 280	sylan2	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to \forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0$
282	281	biantrurd	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to (\exists p \in ran x p (N) \neq 0 \leftrightarrow (\forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (N) \neq 0))$
283		nnuz	$⊢ ℕ = ℤ_{\geq 1}$
284	1 283	eleqtrdi	$⊢ φ \to N \in ℤ_{\geq 1}$
285		fzm1	$⊢ N \in ℤ_{\geq 1} \to (n \in (1 \dots N) \leftrightarrow (n \in (1 \dots N - 1) \lor n = N))$
286	284 285	syl	$⊢ φ \to (n \in (1 \dots N) \leftrightarrow (n \in (1 \dots N - 1) \lor n = N))$
287		elun	$⊢ n \in ((1 \dots N - 1) \cup \{N\}) \leftrightarrow (n \in (1 \dots N - 1) \lor n \in \{N\})$
288	181	orbi2i	$⊢ (n \in (1 \dots N - 1) \lor n \in \{N\}) \leftrightarrow (n \in (1 \dots N - 1) \lor n = N)$
289	287 288	bitri	$⊢ n \in ((1 \dots N - 1) \cup \{N\}) \leftrightarrow (n \in (1 \dots N - 1) \lor n = N)$
290	286 289	bitr4di	$⊢ φ \to (n \in (1 \dots N) \leftrightarrow n \in ((1 \dots N - 1) \cup \{N\}))$
291	290	eqrdv	$⊢ φ \to (1 \dots N) = (1 \dots N - 1) \cup \{N\}$
292	291	raleqdv	$⊢ φ \to (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \leftrightarrow \forall n \in ((1 \dots N - 1) \cup \{N\}) \exists p \in ran x p (n) \neq 0)$
293		ralunb	$⊢ \forall n \in ((1 \dots N - 1) \cup \{N\}) \exists p \in ran x p (n) \neq 0 \leftrightarrow (\forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0 \land \forall n \in \{N\} \exists p \in ran x p (n) \neq 0)$
294	292 293	bitrdi	$⊢ φ \to (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \leftrightarrow (\forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0 \land \forall n \in \{N\} \exists p \in ran x p (n) \neq 0))$
295		fveq2	$⊢ n = N \to p (n) = p (N)$
296	295	neeq1d	$⊢ n = N \to (p (n) \neq 0 \leftrightarrow p (N) \neq 0)$
297	296	rexbidv	$⊢ n = N \to (\exists p \in ran x p (n) \neq 0 \leftrightarrow \exists p \in ran x p (N) \neq 0)$
298	297	ralsng	$⊢ N \in ℕ \to (\forall n \in \{N\} \exists p \in ran x p (n) \neq 0 \leftrightarrow \exists p \in ran x p (N) \neq 0)$
299	1 298	syl	$⊢ φ \to (\forall n \in \{N\} \exists p \in ran x p (n) \neq 0 \leftrightarrow \exists p \in ran x p (N) \neq 0)$
300	299	anbi2d	$⊢ φ \to ((\forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0 \land \forall n \in \{N\} \exists p \in ran x p (n) \neq 0) \leftrightarrow (\forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (N) \neq 0))$
301	294 300	bitrd	$⊢ φ \to (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \leftrightarrow (\forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (N) \neq 0))$
302	301	ad2antrr	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \leftrightarrow (\forall n \in (1 \dots N - 1) \exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (N) \neq 0))$
303		0z	$⊢ 0 \in ℤ$
304		1z	$⊢ 1 \in ℤ$
305		fzshftral	$⊢ (0 \in ℤ \land N - 1 \in ℤ \land 1 \in ℤ) \to (\forall n \in (0 \dots N - 1) \exists p \in ran x n = B \leftrightarrow \forall m \in (0 + 1 \dots N - 1 + 1) \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B)$
306	303 304 305	mp3an13	$⊢ N - 1 \in ℤ \to (\forall n \in (0 \dots N - 1) \exists p \in ran x n = B \leftrightarrow \forall m \in (0 + 1 \dots N - 1 + 1) \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B)$
307	187 248 306	3syl	$⊢ φ \to (\forall n \in (0 \dots N - 1) \exists p \in ran x n = B \leftrightarrow \forall m \in (0 + 1 \dots N - 1 + 1) \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B)$
308		0p1e1	$⊢ 0 + 1 = 1$
309	308	a1i	$⊢ φ \to 0 + 1 = 1$
310	309 247	oveq12d	$⊢ φ \to (0 + 1 \dots N - 1 + 1) = (1 \dots N)$
311	310	raleqdv	$⊢ φ \to (\forall m \in (0 + 1 \dots N - 1 + 1) \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B \leftrightarrow \forall m \in (1 \dots N) \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B)$
312	307 311	bitrd	$⊢ φ \to (\forall n \in (0 \dots N - 1) \exists p \in ran x n = B \leftrightarrow \forall m \in (1 \dots N) \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B)$
313		ovex	$⊢ m - 1 \in V$
314		eqeq1	$⊢ n = m - 1 \to (n = B \leftrightarrow m - 1 = B)$
315	314	rexbidv	$⊢ n = m - 1 \to (\exists p \in ran x n = B \leftrightarrow \exists p \in ran x m - 1 = B)$
316	313 315	sbcie	$⊢ \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B \leftrightarrow \exists p \in ran x m - 1 = B$
317	316	ralbii	$⊢ \forall m \in (1 \dots N) \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B \leftrightarrow \forall m \in (1 \dots N) \exists p \in ran x m - 1 = B$
318		oveq1	$⊢ m = n \to m - 1 = n - 1$
319	318	eqeq1d	$⊢ m = n \to (m - 1 = B \leftrightarrow n - 1 = B)$
320	319	rexbidv	$⊢ m = n \to (\exists p \in ran x m - 1 = B \leftrightarrow \exists p \in ran x n - 1 = B)$
321	320	cbvralvw	$⊢ \forall m \in (1 \dots N) \exists p \in ran x m - 1 = B \leftrightarrow \forall n \in (1 \dots N) \exists p \in ran x n - 1 = B$
322	317 321	bitri	$⊢ \forall m \in (1 \dots N) \underset{˙}{[} (m - 1) / n \underset{˙}{]} \exists p \in ran x n = B \leftrightarrow \forall n \in (1 \dots N) \exists p \in ran x n - 1 = B$
323	312 322	bitrdi	$⊢ φ \to (\forall n \in (0 \dots N - 1) \exists p \in ran x n = B \leftrightarrow \forall n \in (1 \dots N) \exists p \in ran x n - 1 = B)$
324	323	biimpa	$⊢ (φ \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to \forall n \in (1 \dots N) \exists p \in ran x n - 1 = B$
325	324	adantlr	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to \forall n \in (1 \dots N) \exists p \in ran x n - 1 = B$
326	5	necomd	$⊢ (φ \land (n \in (1 \dots N) \land p : (1 \dots N) ⟶ (0 \dots K) \land p (n) = K)) \to n - 1 \neq B$
327	326	3exp2	$⊢ φ \to (n \in (1 \dots N) \to (p : (1 \dots N) ⟶ (0 \dots K) \to (p (n) = K \to n - 1 \neq B)))$
328	327	imp31	$⊢ ((φ \land n \in (1 \dots N)) \land p : (1 \dots N) ⟶ (0 \dots K)) \to (p (n) = K \to n - 1 \neq B)$
329	328	necon2d	$⊢ ((φ \land n \in (1 \dots N)) \land p : (1 \dots N) ⟶ (0 \dots K)) \to (n - 1 = B \to p (n) \neq K)$
330	329	adantllr	$⊢ (((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N)) \land p : (1 \dots N) ⟶ (0 \dots K)) \to (n - 1 = B \to p (n) \neq K)$
331	261 330	syldan	$⊢ (((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N)) \land p \in ran x) \to (n - 1 = B \to p (n) \neq K)$
332	331	reximdva	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land n \in (1 \dots N)) \to (\exists p \in ran x n - 1 = B \to \exists p \in ran x p (n) \neq K)$
333	332	ralimdva	$⊢ (φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \to (\forall n \in (1 \dots N) \exists p \in ran x n - 1 = B \to \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq K)$
334	333	imp	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (1 \dots N) \exists p \in ran x n - 1 = B) \to \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq K$
335	325 334	syldan	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq K$
336	335	biantrud	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \leftrightarrow (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \land \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq K))$
337		r19.26	$⊢ \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K) \leftrightarrow (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \land \forall n \in (1 \dots N) \exists p \in ran x p (n) \neq K)$
338	336 337	bitr4di	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to (\forall n \in (1 \dots N) \exists p \in ran x p (n) \neq 0 \leftrightarrow \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))$
339	282 302 338	3bitr2d	$⊢ ((φ \land ran x \subseteq {(0 \dots K)}^{(1 \dots N)}) \land \forall n \in (0 \dots N - 1) \exists p \in ran x n = B) \to (\exists p \in ran x p (N) \neq 0 \leftrightarrow \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))$
340	233 240 339	syl2an	$⊢ ((φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \land (0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B)) \to (\exists p \in ran x p (N) \neq 0 \leftrightarrow \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))$
341	340	pm5.32da	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to (((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \exists p \in ran x p (N) \neq 0) \leftrightarrow ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))$
342	341	anbi2d	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to ((x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \exists p \in ran x p (N) \neq 0)) \leftrightarrow (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))))$
343	342	rexbidva	$⊢ φ \to (\exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \exists p \in ran x p (N) \neq 0)) \leftrightarrow \exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))))$
344	343	adantr	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to (\exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \exists p \in ran x p (N) \neq 0)) \leftrightarrow \exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))))$
345	195	rexeqdv	$⊢ φ \to (\exists j \in ((0 \dots N) ∖ \{N\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C)$
346	345	biimpd	$⊢ φ \to (\exists j \in ((0 \dots N) ∖ \{N\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C)$
347	346	ralimdv	$⊢ φ \to (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{N\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C)$
348	173	rexeqdv	$⊢ 2^{nd} (t) = N \to (\exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in ((0 \dots N) ∖ \{N\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
349	348	ralbidv	$⊢ 2^{nd} (t) = N \to (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{N\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
350	349	imbi1d	$⊢ 2^{nd} (t) = N \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{N\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C))$
351	347 350	syl5ibrcom	$⊢ φ \to (2^{nd} (t) = N \to (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C))$
352	351	com23	$⊢ φ \to (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to (2^{nd} (t) = N \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C))$
353	352	imp	$⊢ (φ \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (2^{nd} (t) = N \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C)$
354	353	adantrd	$⊢ (φ \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to ((2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C)$
355	354	pm4.71rd	$⊢ (φ \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to ((2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))))$
356		an12	$⊢ (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))) \leftrightarrow (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)))$
357		3anass	$⊢ (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))$
358	357	anbi2i	$⊢ (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \leftrightarrow (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)))$
359	356 358	bitr4i	$⊢ (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))) \leftrightarrow (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))$
360	355 359	bitrdi	$⊢ (φ \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to ((2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \leftrightarrow (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)))$
361	360	notbid	$⊢ (φ \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \leftrightarrow \neg (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)))$
362	361	pm5.32da	$⊢ φ \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))))$
363	362	adantr	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))))$
364	231 344 363	3bitr3d	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to (\exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))))$
365	364	rabbidva	$⊢ φ \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)))\}$
366		iunrab	$⊢ ⋃_{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}} \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \exists x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}$
367		difrab	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)))\}$
368	365 366 367	3eqtr4g	$⊢ φ \to ⋃_{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}} \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}$
369	368	fveq2d	$⊢ φ \to \|⋃_{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}} \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\|$
370	32 33	mp1i	$⊢ (φ \land x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)})) \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} \in Fin$
371		simpl	$⊢ (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))) \to x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))$
372	371	a1i	$⊢ t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to ((x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K))) \to x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))))$
373	372	ss2rabi	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} \subseteq \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}$
374	373	sseli	$⊢ s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} \to s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\}$
375		fveq2	$⊢ t = s \to 2^{nd} (t) = 2^{nd} (s)$
376	375	breq2d	$⊢ t = s \to (y < 2^{nd} (t) \leftrightarrow y < 2^{nd} (s))$
377	376	ifbid	$⊢ t = s \to if (y < 2^{nd} (t), y, y + 1) = if (y < 2^{nd} (s), y, y + 1)$
378	377	csbeq1d	$⊢ t = s \to ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))$
379		2fveq3	$⊢ t = s \to 1^{st} (1^{st} (t)) = 1^{st} (1^{st} (s))$
380		2fveq3	$⊢ t = s \to 2^{nd} (1^{st} (t)) = 2^{nd} (1^{st} (s))$
381	380	imaeq1d	$⊢ t = s \to 2^{nd} (1^{st} (t)) [(1 \dots j)] = 2^{nd} (1^{st} (s)) [(1 \dots j)]$
382	381	xpeq1d	$⊢ t = s \to 2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\} = 2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}$
383	380	imaeq1d	$⊢ t = s \to 2^{nd} (1^{st} (t)) [(j + 1 \dots N)] = 2^{nd} (1^{st} (s)) [(j + 1 \dots N)]$
384	383	xpeq1d	$⊢ t = s \to 2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\} = 2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\}$
385	382 384	uneq12d	$⊢ t = s \to (2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}) = (2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})$
386	379 385	oveq12d	$⊢ t = s \to 1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})) = 1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\}))$
387	386	csbeq2dv	$⊢ t = s \to ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))$
388	378 387	eqtrd	$⊢ t = s \to ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))) = ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))$
389	388	mpteq2dv	$⊢ t = s \to (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\}))))$
390	389	eqeq2d	$⊢ t = s \to (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \leftrightarrow x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))))$
391		eqcom	$⊢ x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) \leftrightarrow (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) = x$
392	390 391	bitrdi	$⊢ t = s \to (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \leftrightarrow (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) = x)$
393	392	elrab	$⊢ s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \leftrightarrow (s \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) = x)$
394	393	simprbi	$⊢ s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\}))))\} \to (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) = x$
395	374 394	syl	$⊢ s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} \to (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) = x$
396	395	rgen	$⊢ \forall s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) = x$
397	396	rgenw	$⊢ \forall x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) \forall s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) = x$
398		invdisj	$⊢ \forall x \in ({({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}) \forall s \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\} (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (s), y, y + 1) / j ⦌ (1^{st} (1^{st} (s)) +_{f} ((2^{nd} (1^{st} (s)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (s)) [(j + 1 \dots N)] \times \{0\})))) = x \to \underset{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}}{Disj} \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}$
399	397 398	mp1i	$⊢ φ \to \underset{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}}{Disj} \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}$
400	13 370 399	hashiun	$⊢ φ \to \|⋃_{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}} \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\| = \sum_{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}} \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|$
401	369 400	eqtr3d	$⊢ φ \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\| = \sum_{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}} \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\|$
402		fo1st	$⊢ 1^{st} : V \underset{onto}{⟶} V$
403		fofun	$⊢ 1^{st} : V \underset{onto}{⟶} V \to Fun 1^{st}$
404	402 403	ax-mp	$⊢ Fun 1^{st}$
405		ssv	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \subseteq V$
406		fof	$⊢ 1^{st} : V \underset{onto}{⟶} V \to 1^{st} : V ⟶ V$
407	402 406	ax-mp	$⊢ 1^{st} : V ⟶ V$
408	407	fdmi	$⊢ dom 1^{st} = V$
409	405 408	sseqtrri	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \subseteq dom 1^{st}$
410		fores	$⊢ (Fun 1^{st} \land \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \subseteq dom 1^{st}) \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}]$
411	404 409 410	mp2an	$⊢ ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}]$
412		fveqeq2	$⊢ t = x \to (2^{nd} (t) = N \leftrightarrow 2^{nd} (x) = N)$
413		fveq2	$⊢ t = x \to 1^{st} (t) = 1^{st} (x)$
414	413	csbeq1d	$⊢ t = x \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ 1^{st} (x) / s ⦌ C$
415	414	eqeq2d	$⊢ t = x \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow i = ⦋ 1^{st} (x) / s ⦌ C)$
416	415	rexbidv	$⊢ t = x \to (\exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C)$
417	416	ralbidv	$⊢ t = x \to (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C)$
418		2fveq3	$⊢ t = x \to 1^{st} (1^{st} (t)) = 1^{st} (1^{st} (x))$
419	418	fveq1d	$⊢ t = x \to 1^{st} (1^{st} (t)) (N) = 1^{st} (1^{st} (x)) (N)$
420	419	eqeq1d	$⊢ t = x \to (1^{st} (1^{st} (t)) (N) = 0 \leftrightarrow 1^{st} (1^{st} (x)) (N) = 0)$
421		2fveq3	$⊢ t = x \to 2^{nd} (1^{st} (t)) = 2^{nd} (1^{st} (x))$
422	421	fveq1d	$⊢ t = x \to 2^{nd} (1^{st} (t)) (N) = 2^{nd} (1^{st} (x)) (N)$
423	422	eqeq1d	$⊢ t = x \to (2^{nd} (1^{st} (t)) (N) = N \leftrightarrow 2^{nd} (1^{st} (x)) (N) = N)$
424	417 420 423	3anbi123d	$⊢ t = x \to ((\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N))$
425	412 424	anbi12d	$⊢ t = x \to ((2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \leftrightarrow (2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)))$
426	425	rexrab	$⊢ \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = s \leftrightarrow \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s)$
427		xp1st	$⊢ x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 1^{st} (x) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})$
428	427	anim1i	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \to (1^{st} (x) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N))$
429		eleq1	$⊢ 1^{st} (x) = s \to (1^{st} (x) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \leftrightarrow s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}))$
430		csbeq1a	$⊢ s = 1^{st} (x) \to C = ⦋ 1^{st} (x) / s ⦌ C$
431	430	eqcoms	$⊢ 1^{st} (x) = s \to C = ⦋ 1^{st} (x) / s ⦌ C$
432	431	eqcomd	$⊢ 1^{st} (x) = s \to ⦋ 1^{st} (x) / s ⦌ C = C$
433	432	eqeq2d	$⊢ 1^{st} (x) = s \to (i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow i = C)$
434	433	rexbidv	$⊢ 1^{st} (x) = s \to (\exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N - 1) i = C)$
435	434	ralbidv	$⊢ 1^{st} (x) = s \to (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C)$
436		fveq2	$⊢ 1^{st} (x) = s \to 1^{st} (1^{st} (x)) = 1^{st} (s)$
437	436	fveq1d	$⊢ 1^{st} (x) = s \to 1^{st} (1^{st} (x)) (N) = 1^{st} (s) (N)$
438	437	eqeq1d	$⊢ 1^{st} (x) = s \to (1^{st} (1^{st} (x)) (N) = 0 \leftrightarrow 1^{st} (s) (N) = 0)$
439		fveq2	$⊢ 1^{st} (x) = s \to 2^{nd} (1^{st} (x)) = 2^{nd} (s)$
440	439	fveq1d	$⊢ 1^{st} (x) = s \to 2^{nd} (1^{st} (x)) (N) = 2^{nd} (s) (N)$
441	440	eqeq1d	$⊢ 1^{st} (x) = s \to (2^{nd} (1^{st} (x)) (N) = N \leftrightarrow 2^{nd} (s) (N) = N)$
442	435 438 441	3anbi123d	$⊢ 1^{st} (x) = s \to ((\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N))$
443	429 442	anbi12d	$⊢ 1^{st} (x) = s \to ((1^{st} (x) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \leftrightarrow (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)))$
444	428 443	syl5ibcom	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \to (1^{st} (x) = s \to (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)))$
445	444	adantrl	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N))) \to (1^{st} (x) = s \to (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)))$
446	445	expimpd	$⊢ x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to (((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s) \to (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)))$
447	446	rexlimiv	$⊢ \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s) \to (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N))$
448		nn0fz0	$⊢ N \in ℕ_{0} \leftrightarrow N \in (0 \dots N)$
449	175 448	sylib	$⊢ φ \to N \in (0 \dots N)$
450		opelxpi	$⊢ (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land N \in (0 \dots N)) \to ⟨s, N⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
451	449 450	sylan2	$⊢ (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land φ) \to ⟨s, N⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
452	451	ancoms	$⊢ (φ \land s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \to ⟨s, N⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
453		opelxp2	$⊢ ⟨s, N⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to N \in (0 \dots N)$
454		op2ndg	$⊢ (s \in V \land N \in (0 \dots N)) \to 2^{nd} (⟨s, N⟩) = N$
455	454	biantrurd	$⊢ (s \in V \land N \in (0 \dots N)) \to ((\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N) \leftrightarrow (2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)))$
456		op1stg	$⊢ (s \in V \land N \in (0 \dots N)) \to 1^{st} (⟨s, N⟩) = s$
457		csbeq1a	$⊢ s = 1^{st} (⟨s, N⟩) \to C = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C$
458	457	eqcoms	$⊢ 1^{st} (⟨s, N⟩) = s \to C = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C$
459	458	eqcomd	$⊢ 1^{st} (⟨s, N⟩) = s \to ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C = C$
460	456 459	syl	$⊢ (s \in V \land N \in (0 \dots N)) \to ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C = C$
461	460	eqeq2d	$⊢ (s \in V \land N \in (0 \dots N)) \to (i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \leftrightarrow i = C)$
462	461	rexbidv	$⊢ (s \in V \land N \in (0 \dots N)) \to (\exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N - 1) i = C)$
463	462	ralbidv	$⊢ (s \in V \land N \in (0 \dots N)) \to (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C)$
464	456	fveq2d	$⊢ (s \in V \land N \in (0 \dots N)) \to 1^{st} (1^{st} (⟨s, N⟩)) = 1^{st} (s)$
465	464	fveq1d	$⊢ (s \in V \land N \in (0 \dots N)) \to 1^{st} (1^{st} (⟨s, N⟩)) (N) = 1^{st} (s) (N)$
466	465	eqeq1d	$⊢ (s \in V \land N \in (0 \dots N)) \to (1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \leftrightarrow 1^{st} (s) (N) = 0)$
467	456	fveq2d	$⊢ (s \in V \land N \in (0 \dots N)) \to 2^{nd} (1^{st} (⟨s, N⟩)) = 2^{nd} (s)$
468	467	fveq1d	$⊢ (s \in V \land N \in (0 \dots N)) \to 2^{nd} (1^{st} (⟨s, N⟩)) (N) = 2^{nd} (s) (N)$
469	468	eqeq1d	$⊢ (s \in V \land N \in (0 \dots N)) \to (2^{nd} (1^{st} (⟨s, N⟩)) (N) = N \leftrightarrow 2^{nd} (s) (N) = N)$
470	463 466 469	3anbi123d	$⊢ (s \in V \land N \in (0 \dots N)) \to ((\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N))$
471	456	biantrud	$⊢ (s \in V \land N \in (0 \dots N)) \to ((2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)) \leftrightarrow ((2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)) \land 1^{st} (⟨s, N⟩) = s))$
472	455 470 471	3bitr3d	$⊢ (s \in V \land N \in (0 \dots N)) \to ((\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N) \leftrightarrow ((2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)) \land 1^{st} (⟨s, N⟩) = s))$
473	49 453 472	sylancr	$⊢ ⟨s, N⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to ((\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N) \leftrightarrow ((2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)) \land 1^{st} (⟨s, N⟩) = s))$
474	473	biimpa	$⊢ (⟨s, N⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)) \to ((2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)) \land 1^{st} (⟨s, N⟩) = s)$
475		fveqeq2	$⊢ x = ⟨s, N⟩ \to (2^{nd} (x) = N \leftrightarrow 2^{nd} (⟨s, N⟩) = N)$
476		fveq2	$⊢ x = ⟨s, N⟩ \to 1^{st} (x) = 1^{st} (⟨s, N⟩)$
477	476	csbeq1d	$⊢ x = ⟨s, N⟩ \to ⦋ 1^{st} (x) / s ⦌ C = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C$
478	477	eqeq2d	$⊢ x = ⟨s, N⟩ \to (i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C)$
479	478	rexbidv	$⊢ x = ⟨s, N⟩ \to (\exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C)$
480	479	ralbidv	$⊢ x = ⟨s, N⟩ \to (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C)$
481		2fveq3	$⊢ x = ⟨s, N⟩ \to 1^{st} (1^{st} (x)) = 1^{st} (1^{st} (⟨s, N⟩))$
482	481	fveq1d	$⊢ x = ⟨s, N⟩ \to 1^{st} (1^{st} (x)) (N) = 1^{st} (1^{st} (⟨s, N⟩)) (N)$
483	482	eqeq1d	$⊢ x = ⟨s, N⟩ \to (1^{st} (1^{st} (x)) (N) = 0 \leftrightarrow 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0)$
484		2fveq3	$⊢ x = ⟨s, N⟩ \to 2^{nd} (1^{st} (x)) = 2^{nd} (1^{st} (⟨s, N⟩))$
485	484	fveq1d	$⊢ x = ⟨s, N⟩ \to 2^{nd} (1^{st} (x)) (N) = 2^{nd} (1^{st} (⟨s, N⟩)) (N)$
486	485	eqeq1d	$⊢ x = ⟨s, N⟩ \to (2^{nd} (1^{st} (x)) (N) = N \leftrightarrow 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)$
487	480 483 486	3anbi123d	$⊢ x = ⟨s, N⟩ \to ((\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N))$
488	475 487	anbi12d	$⊢ x = ⟨s, N⟩ \to ((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \leftrightarrow (2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)))$
489		fveqeq2	$⊢ x = ⟨s, N⟩ \to (1^{st} (x) = s \leftrightarrow 1^{st} (⟨s, N⟩) = s)$
490	488 489	anbi12d	$⊢ x = ⟨s, N⟩ \to (((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s) \leftrightarrow ((2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)) \land 1^{st} (⟨s, N⟩) = s))$
491	490	rspcev	$⊢ (⟨s, N⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land ((2^{nd} (⟨s, N⟩) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (⟨s, N⟩) / s ⦌ C \land 1^{st} (1^{st} (⟨s, N⟩)) (N) = 0 \land 2^{nd} (1^{st} (⟨s, N⟩)) (N) = N)) \land 1^{st} (⟨s, N⟩) = s)) \to \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s)$
492	474 491	syldan	$⊢ (⟨s, N⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)) \to \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s)$
493	452 492	sylan	$⊢ ((φ \land s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)) \to \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s)$
494	493	expl	$⊢ φ \to ((s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)) \to \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s))$
495	447 494	impbid2	$⊢ φ \to (\exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((2^{nd} (x) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (x) / s ⦌ C \land 1^{st} (1^{st} (x)) (N) = 0 \land 2^{nd} (1^{st} (x)) (N) = N)) \land 1^{st} (x) = s) \leftrightarrow (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)))$
496	426 495	bitrid	$⊢ φ \to (\exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = s \leftrightarrow (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)))$
497	496	abbidv	$⊢ φ \to \{s \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = s\} = \{s \| (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N))\}$
498		dfimafn	$⊢ (Fun 1^{st} \land \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \subseteq dom 1^{st}) \to 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}] = \{y \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = y\}$
499	404 409 498	mp2an	$⊢ 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}] = \{y \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = y\}$
500		nfv	$⊢ Ⅎ s 2^{nd} (t) = N$
501		nfcv	$⊢ \underline{Ⅎ} s (0 \dots N - 1)$
502		nfcsb1v	$⊢ \underline{Ⅎ} s ⦋ 1^{st} (t) / s ⦌ C$
503	502	nfeq2	$⊢ Ⅎ s i = ⦋ 1^{st} (t) / s ⦌ C$
504	501 503	nfrexw	$⊢ Ⅎ s \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C$
505	501 504	nfralw	$⊢ Ⅎ s \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C$
506		nfv	$⊢ Ⅎ s 1^{st} (1^{st} (t)) (N) = 0$
507		nfv	$⊢ Ⅎ s 2^{nd} (1^{st} (t)) (N) = N$
508	505 506 507	nf3an	$⊢ Ⅎ s (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)$
509	500 508	nfan	$⊢ Ⅎ s (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))$
510		nfcv	$⊢ \underline{Ⅎ} s ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
511	509 510	nfrabw	$⊢ \underline{Ⅎ} s \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}$
512		nfv	$⊢ Ⅎ s 1^{st} (x) = y$
513	511 512	nfrexw	$⊢ Ⅎ s \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = y$
514		nfv	$⊢ Ⅎ y \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = s$
515		eqeq2	$⊢ y = s \to (1^{st} (x) = y \leftrightarrow 1^{st} (x) = s)$
516	515	rexbidv	$⊢ y = s \to (\exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = y \leftrightarrow \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = s)$
517	513 514 516	cbvabw	$⊢ \{y \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = y\} = \{s \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = s\}$
518	499 517	eqtri	$⊢ 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}] = \{s \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} 1^{st} (x) = s\}$
519		df-rab	$⊢ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} = \{s \| (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N))\}$
520	497 518 519	3eqtr4g	$⊢ φ \to 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}] = \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
521		foeq3	$⊢ 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}] = \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}] \leftrightarrow ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\})$
522	520 521	syl	$⊢ φ \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}] \leftrightarrow ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\})$
523	411 522	mpbii	$⊢ φ \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
524		fof	$⊢ ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} ⟶ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
525	523 524	syl	$⊢ φ \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} ⟶ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
526		fvres	$⊢ x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (x) = 1^{st} (x)$
527		fvres	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (y) = 1^{st} (y)$
528	526 527	eqeqan12d	$⊢ (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}) \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (y) \leftrightarrow 1^{st} (x) = 1^{st} (y))$
529		simpl	$⊢ (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \to 2^{nd} (t) = N$
530	529	a1i	$⊢ t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to ((2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N)) \to 2^{nd} (t) = N)$
531	530	ss2rabi	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \subseteq \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| 2^{nd} (t) = N\}$
532	531	sseli	$⊢ x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \to x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| 2^{nd} (t) = N\}$
533	412	elrab	$⊢ x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| 2^{nd} (t) = N\} \leftrightarrow (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land 2^{nd} (x) = N)$
534	532 533	sylib	$⊢ x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \to (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land 2^{nd} (x) = N)$
535	531	sseli	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \to y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| 2^{nd} (t) = N\}$
536		fveqeq2	$⊢ t = y \to (2^{nd} (t) = N \leftrightarrow 2^{nd} (y) = N)$
537	536	elrab	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| 2^{nd} (t) = N\} \leftrightarrow (y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land 2^{nd} (y) = N)$
538	535 537	sylib	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \to (y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land 2^{nd} (y) = N)$
539		eqtr3	$⊢ (2^{nd} (x) = N \land 2^{nd} (y) = N) \to 2^{nd} (x) = 2^{nd} (y)$
540		xpopth	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((1^{st} (x) = 1^{st} (y) \land 2^{nd} (x) = 2^{nd} (y)) \leftrightarrow x = y)$
541	540	biimpd	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((1^{st} (x) = 1^{st} (y) \land 2^{nd} (x) = 2^{nd} (y)) \to x = y)$
542	541	ancomsd	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((2^{nd} (x) = 2^{nd} (y) \land 1^{st} (x) = 1^{st} (y)) \to x = y)$
543	542	expdimp	$⊢ ((x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \land 2^{nd} (x) = 2^{nd} (y)) \to (1^{st} (x) = 1^{st} (y) \to x = y)$
544	539 543	sylan2	$⊢ ((x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \land (2^{nd} (x) = N \land 2^{nd} (y) = N)) \to (1^{st} (x) = 1^{st} (y) \to x = y)$
545	544	an4s	$⊢ ((x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land 2^{nd} (x) = N) \land (y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land 2^{nd} (y) = N)) \to (1^{st} (x) = 1^{st} (y) \to x = y)$
546	534 538 545	syl2an	$⊢ (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}) \to (1^{st} (x) = 1^{st} (y) \to x = y)$
547	528 546	sylbid	$⊢ (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}) \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (y) \to x = y)$
548	547	rgen2	$⊢ \forall x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \forall y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (y) \to x = y)$
549	525 548	jctir	$⊢ φ \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} ⟶ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \land \forall x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \forall y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (y) \to x = y))$
550		dff13	$⊢ ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{1-1}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \leftrightarrow (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} ⟶ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \land \forall x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \forall y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) (y) \to x = y))$
551	549 550	sylibr	$⊢ φ \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{1-1}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
552		df-f1o	$⊢ ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{1-1 onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \leftrightarrow (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{1-1}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \land ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\})$
553	551 523 552	sylanbrc	$⊢ φ \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{1-1 onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
554		rabfi	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \in Fin$
555	32 554	ax-mp	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \in Fin$
556	555	elexi	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \in V$
557	556	f1oen	$⊢ ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \underset{1-1 onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \approx \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
558	553 557	syl	$⊢ φ \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \approx \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
559		rabfi	$⊢ ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin \to \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \in Fin$
560	29 559	ax-mp	$⊢ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \in Fin$
561		hashen	$⊢ (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \in Fin \land \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\} \in Fin) \to (\|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\| = \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}\| \leftrightarrow \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \approx \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\})$
562	555 560 561	mp2an	$⊢ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\| = \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}\| \leftrightarrow \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\} \approx \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}$
563	558 562	sylibr	$⊢ φ \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\| = \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}\|$
564	563	oveq2d	$⊢ φ \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (2^{nd} (t) = N \land (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = ⦋ 1^{st} (t) / s ⦌ C \land 1^{st} (1^{st} (t)) (N) = 0 \land 2^{nd} (1^{st} (t)) (N) = N))\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}\|$
565	205 401 564	3eqtr3d	$⊢ φ \to \sum_{x \in {({(0 \dots K)}^{(1 \dots N)})}^{(0 \dots N - 1)}} \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (x = (y \in (0 \dots N - 1) ⟼ ⦋ if (y < 2^{nd} (t), y, y + 1) / j ⦌ (1^{st} (1^{st} (t)) +_{f} ((2^{nd} (1^{st} (t)) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (1^{st} (t)) [(j + 1 \dots N)] \times \{0\})))) \land ((0 \dots N - 1) \subseteq ran (p \in ran x ⟼ B) \land \forall n \in (1 \dots N) (\exists p \in ran x p (n) \neq 0 \land \exists p \in ran x p (n) \neq K)))\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}\|$
566	168 565	breqtrd	$⊢ φ \to 2 ∥ (\|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N - 1) i = C \land 1^{st} (s) (N) = 0 \land 2^{nd} (s) (N) = N)\}\|)$

Metamath Proof Explorer

Theorem poimirlem27

Proof