poimirlem26

Description: Lemma for poimir showing an even difference between the number of admissible faces and the number of admissible simplices. Equation (6) 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	$⊢ (φ \land p : (1 \dots N) ⟶ (0 \dots K)) \to B \in (0 \dots N)$
Assertion	poimirlem26	$⊢ φ \to 2 ∥ (\|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}\|)$

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		fzofi	$⊢ (0 ..^K) \in Fin$
5		fzfi	$⊢ (1 \dots N) \in Fin$
6		mapfi	$⊢ ((0 ..^K) \in Fin \land (1 \dots N) \in Fin) \to {(0 ..^K)}^{(1 \dots N)} \in Fin$
7	4 5 6	mp2an	$⊢ {(0 ..^K)}^{(1 \dots N)} \in Fin$
8		mapfi	$⊢ ((1 \dots N) \in Fin \land (1 \dots N) \in Fin) \to {(1 \dots N)}^{(1 \dots N)} \in Fin$
9	5 5 8	mp2an	$⊢ {(1 \dots N)}^{(1 \dots N)} \in Fin$
10		f1of	$⊢ f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N) \to f : (1 \dots N) ⟶ (1 \dots N)$
11	10	ss2abi	$⊢ \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \subseteq \{f \| f : (1 \dots N) ⟶ (1 \dots N)\}$
12		ovex	$⊢ (1 \dots N) \in V$
13	12 12	mapval	$⊢ {(1 \dots N)}^{(1 \dots N)} = \{f \| f : (1 \dots N) ⟶ (1 \dots N)\}$
14	11 13	sseqtrri	$⊢ \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \subseteq {(1 \dots N)}^{(1 \dots N)}$
15		ssfi	$⊢ ({(1 \dots N)}^{(1 \dots N)} \in Fin \land \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \subseteq {(1 \dots N)}^{(1 \dots N)}) \to \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
16	9 14 15	mp2an	$⊢ \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
17	7 16	pm3.2i	$⊢ ({(0 ..^K)}^{(1 \dots N)} \in Fin \land \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin)$
18		xpfi	$⊢ ({(0 ..^K)}^{(1 \dots N)} \in Fin \land \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin) \to ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
19	17 18	mp1i	$⊢ φ \to ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
20		2z	$⊢ 2 \in ℤ$
21	20	a1i	$⊢ φ \to 2 \in ℤ$
22		snfi	$⊢ \{x\} \in Fin$
23		fzfi	$⊢ (0 \dots N) \in Fin$
24		rabfi	$⊢ (0 \dots N) \in Fin \to \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin$
25	23 24	ax-mp	$⊢ \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin$
26		xpfi	$⊢ (\{x\} \in Fin \land \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin) \to \{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin$
27	22 25 26	mp2an	$⊢ \{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin$
28		hashcl	$⊢ \{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin \to \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| \in ℕ_{0}$
29	27 28	ax-mp	$⊢ \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| \in ℕ_{0}$
30	29	nn0zi	$⊢ \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| \in ℤ$
31	30	a1i	$⊢ (φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \to \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| \in ℤ$
32	1	ad2antrr	$⊢ ((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to N \in ℕ$
33		nfv	$⊢ Ⅎ j p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots k)] \times \{1\}) \cup (2^{nd} (t) [(k + 1 \dots N)] \times \{0\}))$
34		nfcsb1v	$⊢ \underline{Ⅎ} j ⦋ k / j ⦌ ⦋ t / s ⦌ C$
35	34	nfeq2	$⊢ Ⅎ j B = ⦋ k / j ⦌ ⦋ t / s ⦌ C$
36	33 35	nfim	$⊢ Ⅎ j (p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots k)] \times \{1\}) \cup (2^{nd} (t) [(k + 1 \dots N)] \times \{0\})) \to B = ⦋ k / j ⦌ ⦋ t / s ⦌ C)$
37		oveq2	$⊢ j = k \to (1 \dots j) = (1 \dots k)$
38	37	imaeq2d	$⊢ j = k \to 2^{nd} (t) [(1 \dots j)] = 2^{nd} (t) [(1 \dots k)]$
39	38	xpeq1d	$⊢ j = k \to 2^{nd} (t) [(1 \dots j)] \times \{1\} = 2^{nd} (t) [(1 \dots k)] \times \{1\}$
40		oveq1	$⊢ j = k \to j + 1 = k + 1$
41	40	oveq1d	$⊢ j = k \to (j + 1 \dots N) = (k + 1 \dots N)$
42	41	imaeq2d	$⊢ j = k \to 2^{nd} (t) [(j + 1 \dots N)] = 2^{nd} (t) [(k + 1 \dots N)]$
43	42	xpeq1d	$⊢ j = k \to 2^{nd} (t) [(j + 1 \dots N)] \times \{0\} = 2^{nd} (t) [(k + 1 \dots N)] \times \{0\}$
44	39 43	uneq12d	$⊢ j = k \to (2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\}) = (2^{nd} (t) [(1 \dots k)] \times \{1\}) \cup (2^{nd} (t) [(k + 1 \dots N)] \times \{0\})$
45	44	oveq2d	$⊢ j = k \to 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\})) = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots k)] \times \{1\}) \cup (2^{nd} (t) [(k + 1 \dots N)] \times \{0\}))$
46	45	eqeq2d	$⊢ j = k \to (p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\})) \leftrightarrow p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots k)] \times \{1\}) \cup (2^{nd} (t) [(k + 1 \dots N)] \times \{0\})))$
47		csbeq1a	$⊢ j = k \to ⦋ t / s ⦌ C = ⦋ k / j ⦌ ⦋ t / s ⦌ C$
48	47	eqeq2d	$⊢ j = k \to (B = ⦋ t / s ⦌ C \leftrightarrow B = ⦋ k / j ⦌ ⦋ t / s ⦌ C)$
49	46 48	imbi12d	$⊢ j = k \to ((p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\})) \to B = ⦋ t / s ⦌ C) \leftrightarrow (p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots k)] \times \{1\}) \cup (2^{nd} (t) [(k + 1 \dots N)] \times \{0\})) \to B = ⦋ k / j ⦌ ⦋ t / s ⦌ C))$
50		nfv	$⊢ Ⅎ s p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\}))$
51		nfcsb1v	$⊢ \underline{Ⅎ} s ⦋ t / s ⦌ C$
52	51	nfeq2	$⊢ Ⅎ s B = ⦋ t / s ⦌ C$
53	50 52	nfim	$⊢ Ⅎ s (p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\})) \to B = ⦋ t / s ⦌ C)$
54		fveq2	$⊢ s = t \to 1^{st} (s) = 1^{st} (t)$
55		fveq2	$⊢ s = t \to 2^{nd} (s) = 2^{nd} (t)$
56	55	imaeq1d	$⊢ s = t \to 2^{nd} (s) [(1 \dots j)] = 2^{nd} (t) [(1 \dots j)]$
57	56	xpeq1d	$⊢ s = t \to 2^{nd} (s) [(1 \dots j)] \times \{1\} = 2^{nd} (t) [(1 \dots j)] \times \{1\}$
58	55	imaeq1d	$⊢ s = t \to 2^{nd} (s) [(j + 1 \dots N)] = 2^{nd} (t) [(j + 1 \dots N)]$
59	58	xpeq1d	$⊢ s = t \to 2^{nd} (s) [(j + 1 \dots N)] \times \{0\} = 2^{nd} (t) [(j + 1 \dots N)] \times \{0\}$
60	57 59	uneq12d	$⊢ s = t \to (2^{nd} (s) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (s) [(j + 1 \dots N)] \times \{0\}) = (2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\})$
61	54 60	oveq12d	$⊢ s = t \to 1^{st} (s) +_{f} ((2^{nd} (s) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (s) [(j + 1 \dots N)] \times \{0\})) = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\}))$
62	61	eqeq2d	$⊢ s = t \to (p = 1^{st} (s) +_{f} ((2^{nd} (s) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (s) [(j + 1 \dots N)] \times \{0\})) \leftrightarrow p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\})))$
63		csbeq1a	$⊢ s = t \to C = ⦋ t / s ⦌ C$
64	63	eqeq2d	$⊢ s = t \to (B = C \leftrightarrow B = ⦋ t / s ⦌ C)$
65	62 64	imbi12d	$⊢ s = t \to ((p = 1^{st} (s) +_{f} ((2^{nd} (s) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (s) [(j + 1 \dots N)] \times \{0\})) \to B = C) \leftrightarrow (p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\})) \to B = ⦋ t / s ⦌ C))$
66	53 65 2	chvarfv	$⊢ p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots j)] \times \{1\}) \cup (2^{nd} (t) [(j + 1 \dots N)] \times \{0\})) \to B = ⦋ t / s ⦌ C$
67	36 49 66	chvarfv	$⊢ p = 1^{st} (t) +_{f} ((2^{nd} (t) [(1 \dots k)] \times \{1\}) \cup (2^{nd} (t) [(k + 1 \dots N)] \times \{0\})) \to B = ⦋ k / j ⦌ ⦋ t / s ⦌ C$
68	3	ad4ant14	$⊢ (((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \land p : (1 \dots N) ⟶ (0 \dots K)) \to B \in (0 \dots N)$
69		xp1st	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to 1^{st} (x) \in ({(0 ..^K)}^{(1 \dots N)})$
70		elmapi	$⊢ 1^{st} (x) \in ({(0 ..^K)}^{(1 \dots N)}) \to 1^{st} (x) : (1 \dots N) ⟶ (0 ..^K)$
71	69 70	syl	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to 1^{st} (x) : (1 \dots N) ⟶ (0 ..^K)$
72	71	ad2antlr	$⊢ ((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to 1^{st} (x) : (1 \dots N) ⟶ (0 ..^K)$
73		xp2nd	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to 2^{nd} (x) \in \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}$
74		fvex	$⊢ 2^{nd} (x) \in V$
75		f1oeq1	$⊢ f = 2^{nd} (x) \to (f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N) \leftrightarrow 2^{nd} (x) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N))$
76	74 75	elab	$⊢ 2^{nd} (x) \in \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \leftrightarrow 2^{nd} (x) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)$
77	73 76	sylib	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to 2^{nd} (x) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)$
78	77	ad2antlr	$⊢ ((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to 2^{nd} (x) : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)$
79		nfcv	$⊢ \underline{Ⅎ} j N$
80		nfcv	$⊢ \underline{Ⅎ} j x$
81	80 34	nfcsbw	$⊢ \underline{Ⅎ} j ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
82	79 81	nfne	$⊢ Ⅎ j N \neq ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
83		nfcv	$⊢ \underline{Ⅎ} t C$
84	83 51 63	cbvcsbw	$⊢ ⦋ x / s ⦌ C = ⦋ x / t ⦌ ⦋ t / s ⦌ C$
85	47	csbeq2dv	$⊢ j = k \to ⦋ x / t ⦌ ⦋ t / s ⦌ C = ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
86	84 85	eqtrid	$⊢ j = k \to ⦋ x / s ⦌ C = ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
87	86	neeq2d	$⊢ j = k \to (N \neq ⦋ x / s ⦌ C \leftrightarrow N \neq ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C)$
88	82 87	rspc	$⊢ k \in (0 \dots N) \to (\forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \to N \neq ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C)$
89	88	impcom	$⊢ (\forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \land k \in (0 \dots N)) \to N \neq ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
90	89	adantll	$⊢ (((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \land k \in (0 \dots N)) \to N \neq ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
91		1st2nd2	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to x = ⟨1^{st} (x), 2^{nd} (x)⟩$
92	91	csbeq1d	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
93	92	ad3antlr	$⊢ (((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \land k \in (0 \dots N)) \to ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
94	90 93	neeqtrd	$⊢ (((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \land k \in (0 \dots N)) \to N \neq ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
95	32 67 68 72 78 94	poimirlem25	$⊢ ((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to 2 ∥ \|\{y \in (0 \dots N) \| \forall i \in (0 \dots N - 1) \exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C\}\|$
96		nfv	$⊢ Ⅎ k i = ⦋ x / s ⦌ C$
97	81	nfeq2	$⊢ Ⅎ j i = ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
98	86	eqeq2d	$⊢ j = k \to (i = ⦋ x / s ⦌ C \leftrightarrow i = ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C)$
99	96 97 98	cbvrexw	$⊢ \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \leftrightarrow \exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C$
100	92	eqeq2d	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to (i = ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C \leftrightarrow i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C)$
101	100	rexbidv	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to (\exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C \leftrightarrow \exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C)$
102	99 101	bitr2id	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to (\exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C \leftrightarrow \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C)$
103	102	ralbidv	$⊢ x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \to (\forall i \in (0 \dots N - 1) \exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C)$
104		iba	$⊢ \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \to (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))$
105	103 104	sylan9bb	$⊢ (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to (\forall i \in (0 \dots N - 1) \exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))$
106	105	rabbidv	$⊢ (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to \{y \in (0 \dots N) \| \forall i \in (0 \dots N - 1) \exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C\} = \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}$
107	106	fveq2d	$⊢ (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to \|\{y \in (0 \dots N) \| \forall i \in (0 \dots N - 1) \exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C\}\| = \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
108	107	adantll	$⊢ ((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to \|\{y \in (0 \dots N) \| \forall i \in (0 \dots N - 1) \exists k \in ((0 \dots N) ∖ \{y\}) i = ⦋ ⟨1^{st} (x), 2^{nd} (x)⟩ / t ⦌ ⦋ k / j ⦌ ⦋ t / s ⦌ C\}\| = \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
109	95 108	breqtrd	$⊢ ((φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to 2 ∥ \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
110	109	ex	$⊢ (φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \to (\forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \to 2 ∥ \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|)$
111		dvds0	$⊢ 2 \in ℤ \to 2 ∥ 0$
112	20 111	ax-mp	$⊢ 2 ∥ 0$
113		hash0	$⊢ \|\emptyset\| = 0$
114	112 113	breqtrri	$⊢ 2 ∥ \|\emptyset\|$
115		simpr	$⊢ (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \to \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C$
116	115	con3i	$⊢ \neg \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \to \neg (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)$
117	116	ralrimivw	$⊢ \neg \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \to \forall y \in (0 \dots N) \neg (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)$
118		rabeq0	$⊢ \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} = \emptyset \leftrightarrow \forall y \in (0 \dots N) \neg (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)$
119	117 118	sylibr	$⊢ \neg \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \to \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} = \emptyset$
120	119	fveq2d	$⊢ \neg \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \to \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| = \|\emptyset\|$
121	114 120	breqtrrid	$⊢ \neg \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \to 2 ∥ \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
122	110 121	pm2.61d1	$⊢ (φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \to 2 ∥ \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
123		hashxp	$⊢ (\{x\} \in Fin \land \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin) \to \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| = \|\{x\}\| \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
124	22 25 123	mp2an	$⊢ \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| = \|\{x\}\| \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
125		vex	$⊢ x \in V$
126		hashsng	$⊢ x \in V \to \|\{x\}\| = 1$
127	125 126	ax-mp	$⊢ \|\{x\}\| = 1$
128	127	oveq1i	$⊢ \|\{x\}\| \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| = 1 \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
129		hashcl	$⊢ \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin \to \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| \in ℕ_{0}$
130	25 129	ax-mp	$⊢ \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| \in ℕ_{0}$
131	130	nn0cni	$⊢ \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| \in ℂ$
132	131	mullidi	$⊢ 1 \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| = \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
133	124 128 132	3eqtri	$⊢ \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| = \|\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
134	122 133	breqtrrdi	$⊢ (φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \to 2 ∥ \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
135	19 21 31 134	fsumdvds	$⊢ φ \to 2 ∥ \sum_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
136	7 16 18	mp2an	$⊢ ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin$
137		xpfi	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin \land (0 \dots N) \in Fin) \to (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin$
138	136 23 137	mp2an	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin$
139		rabfi	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} \in Fin$
140	138 139	ax-mp	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} \in Fin$
141	1	nncnd	$⊢ φ \to N \in ℂ$
142		npcan1	$⊢ N \in ℂ \to N - 1 + 1 = N$
143	141 142	syl	$⊢ φ \to N - 1 + 1 = N$
144		nnm1nn0	$⊢ N \in ℕ \to N - 1 \in ℕ_{0}$
145	1 144	syl	$⊢ φ \to N - 1 \in ℕ_{0}$
146	145	nn0zd	$⊢ φ \to N - 1 \in ℤ$
147		uzid	$⊢ N - 1 \in ℤ \to N - 1 \in ℤ_{\geq (N - 1)}$
148		peano2uz	$⊢ N - 1 \in ℤ_{\geq (N - 1)} \to N - 1 + 1 \in ℤ_{\geq (N - 1)}$
149	146 147 148	3syl	$⊢ φ \to N - 1 + 1 \in ℤ_{\geq (N - 1)}$
150	143 149	eqeltrrd	$⊢ φ \to N \in ℤ_{\geq (N - 1)}$
151		fzss2	$⊢ N \in ℤ_{\geq (N - 1)} \to (0 \dots N - 1) \subseteq (0 \dots N)$
152		ssralv	$⊢ (0 \dots N - 1) \subseteq (0 \dots N) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
153	150 151 152	3syl	$⊢ φ \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
154	153	adantr	$⊢ (φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
155		raldifb	$⊢ \forall j \in (0 \dots N) (j \notin \{2^{nd} (t)\} \to \neg i = ⦋ 1^{st} (t) / s ⦌ C) \leftrightarrow \forall j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \neg i = ⦋ 1^{st} (t) / s ⦌ C$
156		nfv	$⊢ Ⅎ j φ$
157		nfcsb1v	$⊢ \underline{Ⅎ} j ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
158	157	nfeq2	$⊢ Ⅎ j N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
159	156 158	nfan	$⊢ Ⅎ j (φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
160		nfv	$⊢ Ⅎ j i \in (0 \dots N - 1)$
161	159 160	nfan	$⊢ Ⅎ j ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1))$
162		nnel	$⊢ \neg j \notin \{2^{nd} (t)\} \leftrightarrow j \in \{2^{nd} (t)\}$
163		velsn	$⊢ j \in \{2^{nd} (t)\} \leftrightarrow j = 2^{nd} (t)$
164	162 163	bitri	$⊢ \neg j \notin \{2^{nd} (t)\} \leftrightarrow j = 2^{nd} (t)$
165		csbeq1a	$⊢ j = 2^{nd} (t) \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
166	165	eqeq2d	$⊢ j = 2^{nd} (t) \to (N = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
167	166	biimparc	$⊢ (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land j = 2^{nd} (t)) \to N = ⦋ 1^{st} (t) / s ⦌ C$
168	1	nnred	$⊢ φ \to N \in ℝ$
169	168	ltm1d	$⊢ φ \to N - 1 < N$
170	145	nn0red	$⊢ φ \to N - 1 \in ℝ$
171	170 168	ltnled	$⊢ φ \to (N - 1 < N \leftrightarrow \neg N \leq N - 1)$
172	169 171	mpbid	$⊢ φ \to \neg N \leq N - 1$
173		elfzle2	$⊢ N \in (0 \dots N - 1) \to N \leq N - 1$
174	172 173	nsyl	$⊢ φ \to \neg N \in (0 \dots N - 1)$
175		eleq1	$⊢ i = N \to (i \in (0 \dots N - 1) \leftrightarrow N \in (0 \dots N - 1))$
176	175	notbid	$⊢ i = N \to (\neg i \in (0 \dots N - 1) \leftrightarrow \neg N \in (0 \dots N - 1))$
177	174 176	syl5ibrcom	$⊢ φ \to (i = N \to \neg i \in (0 \dots N - 1))$
178	177	con2d	$⊢ φ \to (i \in (0 \dots N - 1) \to \neg i = N)$
179	178	imp	$⊢ (φ \land i \in (0 \dots N - 1)) \to \neg i = N$
180		eqeq2	$⊢ N = ⦋ 1^{st} (t) / s ⦌ C \to (i = N \leftrightarrow i = ⦋ 1^{st} (t) / s ⦌ C)$
181	180	notbid	$⊢ N = ⦋ 1^{st} (t) / s ⦌ C \to (\neg i = N \leftrightarrow \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
182	179 181	syl5ibcom	$⊢ (φ \land i \in (0 \dots N - 1)) \to (N = ⦋ 1^{st} (t) / s ⦌ C \to \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
183	167 182	syl5	$⊢ (φ \land i \in (0 \dots N - 1)) \to ((N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land j = 2^{nd} (t)) \to \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
184	183	expdimp	$⊢ ((φ \land i \in (0 \dots N - 1)) \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \to (j = 2^{nd} (t) \to \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
185	184	an32s	$⊢ ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \to (j = 2^{nd} (t) \to \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
186	164 185	biimtrid	$⊢ ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \to (\neg j \notin \{2^{nd} (t)\} \to \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
187		idd	$⊢ ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \to (\neg i = ⦋ 1^{st} (t) / s ⦌ C \to \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
188	186 187	jad	$⊢ ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \to ((j \notin \{2^{nd} (t)\} \to \neg i = ⦋ 1^{st} (t) / s ⦌ C) \to \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
189	188	adantr	$⊢ (((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \land j \in (0 \dots N)) \to ((j \notin \{2^{nd} (t)\} \to \neg i = ⦋ 1^{st} (t) / s ⦌ C) \to \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
190	161 189	ralimdaa	$⊢ ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \to (\forall j \in (0 \dots N) (j \notin \{2^{nd} (t)\} \to \neg i = ⦋ 1^{st} (t) / s ⦌ C) \to \forall j \in (0 \dots N) \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
191	155 190	biimtrrid	$⊢ ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \to (\forall j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \neg i = ⦋ 1^{st} (t) / s ⦌ C \to \forall j \in (0 \dots N) \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
192	191	con3d	$⊢ ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \to (\neg \forall j \in (0 \dots N) \neg i = ⦋ 1^{st} (t) / s ⦌ C \to \neg \forall j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \neg i = ⦋ 1^{st} (t) / s ⦌ C)$
193		dfrex2	$⊢ \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \neg \forall j \in (0 \dots N) \neg i = ⦋ 1^{st} (t) / s ⦌ C$
194		dfrex2	$⊢ \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \neg \forall j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \neg i = ⦋ 1^{st} (t) / s ⦌ C$
195	192 193 194	3imtr4g	$⊢ ((φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \land i \in (0 \dots N - 1)) \to (\exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \to \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
196	195	ralimdva	$⊢ (φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \to (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
197	154 196	syld	$⊢ (φ \land N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
198	197	expimpd	$⊢ φ \to ((N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C) \to \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
199	198	adantr	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C) \to \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
200	199	ss2rabdv	$⊢ φ \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \subseteq \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}$
201		hashssdif	$⊢ (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} \in Fin \land \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \subseteq \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}) \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\|$
202	140 200 201	sylancr	$⊢ φ \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\|$
203		xp2nd	$⊢ t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 2^{nd} (t) \in (0 \dots N)$
204		df-ne	$⊢ N \neq ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \neg N = ⦋ 1^{st} (t) / s ⦌ C$
205	204	ralbii	$⊢ \forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall j \in (0 \dots N) \neg N = ⦋ 1^{st} (t) / s ⦌ C$
206		ralnex	$⊢ \forall j \in (0 \dots N) \neg N = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \neg \exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C$
207	205 206	bitri	$⊢ \forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \neg \exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C$
208	1	nnnn0d	$⊢ φ \to N \in ℕ_{0}$
209		nn0uz	$⊢ ℕ_{0} = ℤ_{\geq 0}$
210	208 209	eleqtrdi	$⊢ φ \to N \in ℤ_{\geq 0}$
211	143 210	eqeltrd	$⊢ φ \to N - 1 + 1 \in ℤ_{\geq 0}$
212		fzsplit2	$⊢ (N - 1 + 1 \in ℤ_{\geq 0} \land N \in ℤ_{\geq (N - 1)}) \to (0 \dots N) = (0 \dots N - 1) \cup (N - 1 + 1 \dots N)$
213	211 150 212	syl2anc	$⊢ φ \to (0 \dots N) = (0 \dots N - 1) \cup (N - 1 + 1 \dots N)$
214	143	oveq1d	$⊢ φ \to (N - 1 + 1 \dots N) = (N \dots N)$
215	1	nnzd	$⊢ φ \to N \in ℤ$
216		fzsn	$⊢ N \in ℤ \to (N \dots N) = \{N\}$
217	215 216	syl	$⊢ φ \to (N \dots N) = \{N\}$
218	214 217	eqtrd	$⊢ φ \to (N - 1 + 1 \dots N) = \{N\}$
219	218	uneq2d	$⊢ φ \to (0 \dots N - 1) \cup (N - 1 + 1 \dots N) = (0 \dots N - 1) \cup \{N\}$
220	213 219	eqtrd	$⊢ φ \to (0 \dots N) = (0 \dots N - 1) \cup \{N\}$
221	220	raleqdv	$⊢ φ \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in ((0 \dots N - 1) \cup \{N\}) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
222		ralunb	$⊢ \forall i \in ((0 \dots N - 1) \cup \{N\}) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
223		difss	$⊢ (0 \dots N) ∖ \{2^{nd} (t)\} \subseteq (0 \dots N)$
224		ssrexv	$⊢ (0 \dots N) ∖ \{2^{nd} (t)\} \subseteq (0 \dots N) \to (\exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
225	223 224	ax-mp	$⊢ \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C$
226	225	ralimi	$⊢ \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C$
227	226	biantrurd	$⊢ \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to (\forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C))$
228	222 227	bitr4id	$⊢ \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to (\forall i \in ((0 \dots N - 1) \cup \{N\}) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
229	221 228	sylan9bb	$⊢ (φ \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
230	229	adantlr	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
231		nn0fz0	$⊢ N \in ℕ_{0} \leftrightarrow N \in (0 \dots N)$
232	208 231	sylib	$⊢ φ \to N \in (0 \dots N)$
233	232	ad2antrr	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to N \in (0 \dots N)$
234		eqeq1	$⊢ i = N \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow N = ⦋ 1^{st} (t) / s ⦌ C)$
235	234	rexbidv	$⊢ i = N \to (\exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C)$
236	235	rspcva	$⊢ (N \in (0 \dots N) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C) \to \exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C$
237		nfv	$⊢ Ⅎ j (φ \land 2^{nd} (t) \in (0 \dots N))$
238		nfcv	$⊢ \underline{Ⅎ} j (0 \dots N - 1)$
239		nfre1	$⊢ Ⅎ j \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C$
240	238 239	nfralw	$⊢ Ⅎ j \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C$
241	237 240	nfan	$⊢ Ⅎ j ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C)$
242		eleq1	$⊢ N = ⦋ 1^{st} (t) / s ⦌ C \to (N \in (0 \dots N - 1) \leftrightarrow ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
243	242	notbid	$⊢ N = ⦋ 1^{st} (t) / s ⦌ C \to (\neg N \in (0 \dots N - 1) \leftrightarrow \neg ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
244	174 243	syl5ibcom	$⊢ φ \to (N = ⦋ 1^{st} (t) / s ⦌ C \to \neg ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
245	244	ad3antrrr	$⊢ (((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in (0 \dots N)) \to (N = ⦋ 1^{st} (t) / s ⦌ C \to \neg ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
246		eldifsn	$⊢ j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \leftrightarrow (j \in (0 \dots N) \land j \neq 2^{nd} (t))$
247		diffi	$⊢ (0 \dots N) \in Fin \to (0 \dots N) ∖ \{2^{nd} (t)\} \in Fin$
248	23 247	ax-mp	$⊢ (0 \dots N) ∖ \{2^{nd} (t)\} \in Fin$
249		ssrab2	$⊢ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \subseteq (0 \dots N) ∖ \{2^{nd} (t)\}$
250		ssdomg	$⊢ (0 \dots N) ∖ \{2^{nd} (t)\} \in Fin \to (\{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \subseteq (0 \dots N) ∖ \{2^{nd} (t)\} \to \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ≼ ((0 \dots N) ∖ \{2^{nd} (t)\}))$
251	248 249 250	mp2	$⊢ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ≼ ((0 \dots N) ∖ \{2^{nd} (t)\})$
252		hashdifsn	$⊢ ((0 \dots N) \in Fin \land 2^{nd} (t) \in (0 \dots N)) \to \|(0 \dots N) ∖ \{2^{nd} (t)\}\| = \|(0 \dots N)\| - 1$
253	23 252	mpan	$⊢ 2^{nd} (t) \in (0 \dots N) \to \|(0 \dots N) ∖ \{2^{nd} (t)\}\| = \|(0 \dots N)\| - 1$
254		1cnd	$⊢ φ \to 1 \in ℂ$
255	141 254 254	addsubd	$⊢ φ \to N + 1 - 1 = N - 1 + 1$
256		hashfz0	$⊢ N \in ℕ_{0} \to \|(0 \dots N)\| = N + 1$
257	208 256	syl	$⊢ φ \to \|(0 \dots N)\| = N + 1$
258	257	oveq1d	$⊢ φ \to \|(0 \dots N)\| - 1 = N + 1 - 1$
259		hashfz0	$⊢ N - 1 \in ℕ_{0} \to \|(0 \dots N - 1)\| = N - 1 + 1$
260	145 259	syl	$⊢ φ \to \|(0 \dots N - 1)\| = N - 1 + 1$
261	255 258 260	3eqtr4d	$⊢ φ \to \|(0 \dots N)\| - 1 = \|(0 \dots N - 1)\|$
262	253 261	sylan9eqr	$⊢ (φ \land 2^{nd} (t) \in (0 \dots N)) \to \|(0 \dots N) ∖ \{2^{nd} (t)\}\| = \|(0 \dots N - 1)\|$
263		fzfi	$⊢ (0 \dots N - 1) \in Fin$
264		hashen	$⊢ ((0 \dots N) ∖ \{2^{nd} (t)\} \in Fin \land (0 \dots N - 1) \in Fin) \to (\|(0 \dots N) ∖ \{2^{nd} (t)\}\| = \|(0 \dots N - 1)\| \leftrightarrow ((0 \dots N) ∖ \{2^{nd} (t)\}) \approx (0 \dots N - 1))$
265	248 263 264	mp2an	$⊢ \|(0 \dots N) ∖ \{2^{nd} (t)\}\| = \|(0 \dots N - 1)\| \leftrightarrow ((0 \dots N) ∖ \{2^{nd} (t)\}) \approx (0 \dots N - 1)$
266	262 265	sylib	$⊢ (φ \land 2^{nd} (t) \in (0 \dots N)) \to ((0 \dots N) ∖ \{2^{nd} (t)\}) \approx (0 \dots N - 1)$
267		rabfi	$⊢ (0 \dots N) ∖ \{2^{nd} (t)\} \in Fin \to \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \in Fin$
268	248 267	ax-mp	$⊢ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \in Fin$
269		eleq1	$⊢ i = ⦋ 1^{st} (t) / s ⦌ C \to (i \in (0 \dots N - 1) \leftrightarrow ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
270	269	biimpac	$⊢ (i \in (0 \dots N - 1) \land i = ⦋ 1^{st} (t) / s ⦌ C) \to ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)$
271		rabid	$⊢ j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \leftrightarrow (j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \land ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
272	271	simplbi2com	$⊢ ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1) \to (j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \to j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\})$
273	270 272	syl	$⊢ (i \in (0 \dots N - 1) \land i = ⦋ 1^{st} (t) / s ⦌ C) \to (j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \to j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\})$
274	273	impancom	$⊢ (i \in (0 \dots N - 1) \land j \in ((0 \dots N) ∖ \{2^{nd} (t)\})) \to (i = ⦋ 1^{st} (t) / s ⦌ C \to j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\})$
275	274	ancrd	$⊢ (i \in (0 \dots N - 1) \land j \in ((0 \dots N) ∖ \{2^{nd} (t)\})) \to (i = ⦋ 1^{st} (t) / s ⦌ C \to (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C))$
276	275	expimpd	$⊢ i \in (0 \dots N - 1) \to ((j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \land i = ⦋ 1^{st} (t) / s ⦌ C) \to (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C))$
277	276	reximdv2	$⊢ i \in (0 \dots N - 1) \to (\exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \exists j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} i = ⦋ 1^{st} (t) / s ⦌ C)$
278	271	simplbi	$⊢ j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \to j \in ((0 \dots N) ∖ \{2^{nd} (t)\})$
279	274	pm4.71rd	$⊢ (i \in (0 \dots N - 1) \land j \in ((0 \dots N) ∖ \{2^{nd} (t)\})) \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C))$
280		df-mpt	$⊢ (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) = \{⟨k, i⟩ \| (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)\}$
281		nfv	$⊢ Ⅎ k (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)$
282		nfrab1	$⊢ \underline{Ⅎ} j \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}$
283	282	nfcri	$⊢ Ⅎ j k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}$
284		nfcsb1v	$⊢ \underline{Ⅎ} j ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
285	284	nfeq2	$⊢ Ⅎ j i = ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
286	283 285	nfan	$⊢ Ⅎ j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
287		eleq1	$⊢ j = k \to (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \leftrightarrow k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\})$
288		csbeq1a	$⊢ j = k \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
289	288	eqeq2d	$⊢ j = k \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow i = ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
290	287 289	anbi12d	$⊢ j = k \to ((j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C) \leftrightarrow (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C))$
291	281 286 290	cbvopab1	$⊢ \{⟨j, i⟩ \| (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)\} = \{⟨k, i⟩ \| (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)\}$
292	280 291	eqtr4i	$⊢ (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) = \{⟨j, i⟩ \| (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)\}$
293	292	breqi	$⊢ j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i \leftrightarrow j \{⟨j, i⟩ \| (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)\} i$
294		df-br	$⊢ j \{⟨j, i⟩ \| (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)\} i \leftrightarrow ⟨j, i⟩ \in \{⟨j, i⟩ \| (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)\}$
295		opabidw	$⊢ ⟨j, i⟩ \in \{⟨j, i⟩ \| (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)\} \leftrightarrow (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)$
296	293 294 295	3bitri	$⊢ j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i \leftrightarrow (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \land i = ⦋ 1^{st} (t) / s ⦌ C)$
297	279 296	bitr4di	$⊢ (i \in (0 \dots N - 1) \land j \in ((0 \dots N) ∖ \{2^{nd} (t)\})) \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i)$
298	278 297	sylan2	$⊢ (i \in (0 \dots N - 1) \land j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}) \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i)$
299	298	rexbidva	$⊢ i \in (0 \dots N - 1) \to (\exists j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i)$
300		nfcv	$⊢ \underline{Ⅎ} p \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}$
301		nfv	$⊢ Ⅎ p j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i$
302		nfcv	$⊢ \underline{Ⅎ} j p$
303	282 284	nfmpt	$⊢ \underline{Ⅎ} j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
304		nfcv	$⊢ \underline{Ⅎ} j i$
305	302 303 304	nfbr	$⊢ Ⅎ j p (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i$
306		breq1	$⊢ j = p \to (j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i \leftrightarrow p (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i)$
307	282 300 301 305 306	cbvrexfw	$⊢ \exists j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} j (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i \leftrightarrow \exists p \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} p (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i$
308	299 307	bitrdi	$⊢ i \in (0 \dots N - 1) \to (\exists j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists p \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} p (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i)$
309	277 308	sylibd	$⊢ i \in (0 \dots N - 1) \to (\exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \exists p \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} p (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i)$
310	309	ralimia	$⊢ \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to \forall i \in (0 \dots N - 1) \exists p \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} p (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i$
311		eqid	$⊢ (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) = (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
312		nfcv	$⊢ \underline{Ⅎ} j k$
313		nfcv	$⊢ \underline{Ⅎ} j ((0 \dots N) ∖ \{2^{nd} (t)\})$
314	284	nfel1	$⊢ Ⅎ j ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)$
315	288	eleq1d	$⊢ j = k \to (⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1) \leftrightarrow ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
316	312 313 314 315	elrabf	$⊢ k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \leftrightarrow (k \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \land ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
317	316	simprbi	$⊢ k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \to ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)$
318	311 317	fmpti	$⊢ (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) : \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟶ (0 \dots N - 1)$
319	310 318	jctil	$⊢ \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to ((k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) : \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟶ (0 \dots N - 1) \land \forall i \in (0 \dots N - 1) \exists p \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} p (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i)$
320		dffo4	$⊢ (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) : \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \underset{onto}{⟶} (0 \dots N - 1) \leftrightarrow ((k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) : \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟶ (0 \dots N - 1) \land \forall i \in (0 \dots N - 1) \exists p \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} p (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) i)$
321	319 320	sylibr	$⊢ \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) : \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \underset{onto}{⟶} (0 \dots N - 1)$
322		fodomfi	$⊢ (\{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \in Fin \land (k \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ⟼ ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) : \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \underset{onto}{⟶} (0 \dots N - 1)) \to (0 \dots N - 1) ≼ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}$
323	268 321 322	sylancr	$⊢ \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \to (0 \dots N - 1) ≼ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}$
324		endomtr	$⊢ (((0 \dots N) ∖ \{2^{nd} (t)\}) \approx (0 \dots N - 1) \land (0 \dots N - 1) ≼ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}) \to ((0 \dots N) ∖ \{2^{nd} (t)\}) ≼ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}$
325	266 323 324	syl2an	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to ((0 \dots N) ∖ \{2^{nd} (t)\}) ≼ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}$
326		sbth	$⊢ (\{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} ≼ ((0 \dots N) ∖ \{2^{nd} (t)\}) \land ((0 \dots N) ∖ \{2^{nd} (t)\}) ≼ \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}) \to \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \approx ((0 \dots N) ∖ \{2^{nd} (t)\})$
327	251 325 326	sylancr	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \approx ((0 \dots N) ∖ \{2^{nd} (t)\})$
328		fisseneq	$⊢ ((0 \dots N) ∖ \{2^{nd} (t)\} \in Fin \land \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \subseteq (0 \dots N) ∖ \{2^{nd} (t)\} \land \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \approx ((0 \dots N) ∖ \{2^{nd} (t)\})) \to \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} = (0 \dots N) ∖ \{2^{nd} (t)\}$
329	248 249 327 328	mp3an12i	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} = (0 \dots N) ∖ \{2^{nd} (t)\}$
330	329	eleq2d	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \leftrightarrow j \in ((0 \dots N) ∖ \{2^{nd} (t)\}))$
331	330	biimpar	$⊢ (((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in ((0 \dots N) ∖ \{2^{nd} (t)\})) \to j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\}$
332	288	equcoms	$⊢ k = j \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
333	332	eqcomd	$⊢ k = j \to ⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C = ⦋ 1^{st} (t) / s ⦌ C$
334	333	eleq1d	$⊢ k = j \to (⦋ k / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1) \leftrightarrow ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
335	334 317	vtoclga	$⊢ j \in \{j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) \| ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)\} \to ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)$
336	331 335	syl	$⊢ (((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in ((0 \dots N) ∖ \{2^{nd} (t)\})) \to ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)$
337	246 336	sylan2br	$⊢ (((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land (j \in (0 \dots N) \land j \neq 2^{nd} (t))) \to ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1)$
338	337	expr	$⊢ (((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in (0 \dots N)) \to (j \neq 2^{nd} (t) \to ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1))$
339	338	necon1bd	$⊢ (((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in (0 \dots N)) \to (\neg ⦋ 1^{st} (t) / s ⦌ C \in (0 \dots N - 1) \to j = 2^{nd} (t))$
340	245 339	syld	$⊢ (((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in (0 \dots N)) \to (N = ⦋ 1^{st} (t) / s ⦌ C \to j = 2^{nd} (t))$
341	340	imp	$⊢ ((((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in (0 \dots N)) \land N = ⦋ 1^{st} (t) / s ⦌ C) \to j = 2^{nd} (t)$
342	341 165	syl	$⊢ ((((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in (0 \dots N)) \land N = ⦋ 1^{st} (t) / s ⦌ C) \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
343		eqtr	$⊢ (N = ⦋ 1^{st} (t) / s ⦌ C \land ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C) \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
344	343	ex	$⊢ N = ⦋ 1^{st} (t) / s ⦌ C \to (⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
345	344	adantl	$⊢ ((((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in (0 \dots N)) \land N = ⦋ 1^{st} (t) / s ⦌ C) \to (⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
346	342 345	mpd	$⊢ ((((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \land j \in (0 \dots N)) \land N = ⦋ 1^{st} (t) / s ⦌ C) \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
347	346	exp31	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (j \in (0 \dots N) \to (N = ⦋ 1^{st} (t) / s ⦌ C \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C))$
348	241 158 347	rexlimd	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
349	236 348	syl5	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to ((N \in (0 \dots N) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C) \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
350	233 349	mpand	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
351	350	pm4.71rd	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C))$
352	235	ralsng	$⊢ N \in ℕ \to (\forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C)$
353	1 352	syl	$⊢ φ \to (\forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C)$
354	353	ad2antrr	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\forall i \in \{N\} \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C)$
355	230 351 354	3bitr3rd	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C))$
356	355	notbid	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\neg \exists j \in (0 \dots N) N = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \neg (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C))$
357	207 356	bitrid	$⊢ ((φ \land 2^{nd} (t) \in (0 \dots N)) \land \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C) \to (\forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \neg (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C))$
358	357	pm5.32da	$⊢ (φ \land 2^{nd} (t) \in (0 \dots N)) \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)))$
359	203 358	sylan2	$⊢ (φ \land t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)))$
360	359	rabbidva	$⊢ φ \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C)\} = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C))\}$
361		nfv	$⊢ Ⅎ y t = ⟨x, k⟩$
362		nfv	$⊢ Ⅎ y x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})$
363		nfrab1	$⊢ \underline{Ⅎ} y \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}$
364	363	nfcri	$⊢ Ⅎ y k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}$
365	362 364	nfan	$⊢ Ⅎ y (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})$
366	361 365	nfan	$⊢ Ⅎ y (t = ⟨x, k⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}))$
367		nfv	$⊢ Ⅎ k (t = ⟨x, y⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)))$
368		opeq2	$⊢ k = y \to ⟨x, k⟩ = ⟨x, y⟩$
369	368	eqeq2d	$⊢ k = y \to (t = ⟨x, k⟩ \leftrightarrow t = ⟨x, y⟩)$
370		eleq1	$⊢ k = y \to (k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \leftrightarrow y \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})$
371		rabid	$⊢ y \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \leftrightarrow (y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))$
372	370 371	bitrdi	$⊢ k = y \to (k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \leftrightarrow (y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)))$
373	372	anbi2d	$⊢ k = y \to ((x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \leftrightarrow (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))))$
374		3anass	$⊢ (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)) \leftrightarrow (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)))$
375	373 374	bitr4di	$⊢ k = y \to ((x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \leftrightarrow (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)))$
376	369 375	anbi12d	$⊢ k = y \to ((t = ⟨x, k⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})) \leftrightarrow (t = ⟨x, y⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))))$
377	366 367 376	cbvexv1	$⊢ \exists k (t = ⟨x, k⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})) \leftrightarrow \exists y (t = ⟨x, y⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)))$
378	377	exbii	$⊢ \exists x \exists k (t = ⟨x, k⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})) \leftrightarrow \exists x \exists y (t = ⟨x, y⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)))$
379		eliunxp	$⊢ t \in ⋃_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \leftrightarrow \exists x \exists k (t = ⟨x, k⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land k \in \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}))$
380		elopab	$⊢ t \in \{⟨x, y⟩ \| (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))\} \leftrightarrow \exists x \exists y (t = ⟨x, y⟩ \land (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)))$
381	378 379 380	3bitr4i	$⊢ t \in ⋃_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \leftrightarrow t \in \{⟨x, y⟩ \| (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))\}$
382	381	eqriv	$⊢ ⋃_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) = \{⟨x, y⟩ \| (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))\}$
383		vex	$⊢ y \in V$
384	125 383	op2ndd	$⊢ t = ⟨x, y⟩ \to 2^{nd} (t) = y$
385	384	sneqd	$⊢ t = ⟨x, y⟩ \to \{2^{nd} (t)\} = \{y\}$
386	385	difeq2d	$⊢ t = ⟨x, y⟩ \to (0 \dots N) ∖ \{2^{nd} (t)\} = (0 \dots N) ∖ \{y\}$
387	125 383	op1std	$⊢ t = ⟨x, y⟩ \to 1^{st} (t) = x$
388	387	csbeq1d	$⊢ t = ⟨x, y⟩ \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ x / s ⦌ C$
389	388	eqeq2d	$⊢ t = ⟨x, y⟩ \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow i = ⦋ x / s ⦌ C)$
390	386 389	rexeqbidv	$⊢ t = ⟨x, y⟩ \to (\exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C)$
391	390	ralbidv	$⊢ t = ⟨x, y⟩ \to (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C)$
392	388	neeq2d	$⊢ t = ⟨x, y⟩ \to (N \neq ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow N \neq ⦋ x / s ⦌ C)$
393	392	ralbidv	$⊢ t = ⟨x, y⟩ \to (\forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)$
394	391 393	anbi12d	$⊢ t = ⟨x, y⟩ \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))$
395	394	rabxp	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C)\} = \{⟨x, y⟩ \| (x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land y \in (0 \dots N) \land (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C))\}$
396	382 395	eqtr4i	$⊢ ⋃_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ 1^{st} (t) / s ⦌ C)\}$
397		difrab	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C \land \neg (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C))\}$
398	360 396 397	3eqtr4g	$⊢ φ \to ⋃_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) = \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}$
399	398	fveq2d	$⊢ φ \to \|⋃_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\|$
400	27	a1i	$⊢ (φ \land x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \to \{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \in Fin$
401		inxp	$⊢ (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \cap (\{t\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}) = (\{x\} \cap \{t\}) \times (\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \cap \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\})$
402		df-ne	$⊢ x \neq t \leftrightarrow \neg x = t$
403		disjsn2	$⊢ x \neq t \to \{x\} \cap \{t\} = \emptyset$
404	402 403	sylbir	$⊢ \neg x = t \to \{x\} \cap \{t\} = \emptyset$
405	404	xpeq1d	$⊢ \neg x = t \to (\{x\} \cap \{t\}) \times (\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \cap \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}) = \emptyset \times (\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \cap \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\})$
406		0xp	$⊢ \emptyset \times (\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \cap \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}) = \emptyset$
407	405 406	eqtrdi	$⊢ \neg x = t \to (\{x\} \cap \{t\}) \times (\{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} \cap \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}) = \emptyset$
408	401 407	eqtrid	$⊢ \neg x = t \to (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \cap (\{t\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}) = \emptyset$
409	408	orri	$⊢ (x = t \lor (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \cap (\{t\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}) = \emptyset)$
410	409	rgen2w	$⊢ \forall x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \forall t \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) (x = t \lor (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \cap (\{t\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}) = \emptyset)$
411		sneq	$⊢ x = t \to \{x\} = \{t\}$
412		csbeq1	$⊢ x = t \to ⦋ x / s ⦌ C = ⦋ t / s ⦌ C$
413	412	eqeq2d	$⊢ x = t \to (i = ⦋ x / s ⦌ C \leftrightarrow i = ⦋ t / s ⦌ C)$
414	413	rexbidv	$⊢ x = t \to (\exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \leftrightarrow \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C)$
415	414	ralbidv	$⊢ x = t \to (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \leftrightarrow \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C)$
416	412	neeq2d	$⊢ x = t \to (N \neq ⦋ x / s ⦌ C \leftrightarrow N \neq ⦋ t / s ⦌ C)$
417	416	ralbidv	$⊢ x = t \to (\forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C \leftrightarrow \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)$
418	415 417	anbi12d	$⊢ x = t \to ((\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C) \leftrightarrow (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C))$
419	418	rabbidv	$⊢ x = t \to \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} = \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}$
420	411 419	xpeq12d	$⊢ x = t \to \{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\} = \{t\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}$
421	420	disjor	$⊢ \underset{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}}{Disj} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \leftrightarrow \forall x \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \forall t \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) (x = t \lor (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}) \cap (\{t\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ t / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ t / s ⦌ C)\}) = \emptyset)$
422	410 421	mpbir	$⊢ \underset{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}}{Disj} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})$
423	422	a1i	$⊢ φ \to \underset{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}}{Disj} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})$
424	19 400 423	hashiun	$⊢ φ \to \|⋃_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} (\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\})\| = \sum_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
425	399 424	eqtr3d	$⊢ φ \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\} ∖ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\| = \sum_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\|$
426		fo1st	$⊢ 1^{st} : V \underset{onto}{⟶} V$
427		fofun	$⊢ 1^{st} : V \underset{onto}{⟶} V \to Fun 1^{st}$
428	426 427	ax-mp	$⊢ Fun 1^{st}$
429		ssv	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \subseteq V$
430		fof	$⊢ 1^{st} : V \underset{onto}{⟶} V \to 1^{st} : V ⟶ V$
431	426 430	ax-mp	$⊢ 1^{st} : V ⟶ V$
432	431	fdmi	$⊢ dom 1^{st} = V$
433	429 432	sseqtrri	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \subseteq dom 1^{st}$
434		fores	$⊢ (Fun 1^{st} \land \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \subseteq dom 1^{st}) \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{onto}{⟶} 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}]$
435	428 433 434	mp2an	$⊢ ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{onto}{⟶} 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}]$
436		fveq2	$⊢ t = x \to 2^{nd} (t) = 2^{nd} (x)$
437	436	csbeq1d	$⊢ t = x \to ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
438		fveq2	$⊢ t = x \to 1^{st} (t) = 1^{st} (x)$
439	438	csbeq1d	$⊢ t = x \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ 1^{st} (x) / s ⦌ C$
440	439	csbeq2dv	$⊢ t = x \to ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
441	437 440	eqtrd	$⊢ t = x \to ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
442	441	eqeq2d	$⊢ t = x \to (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C)$
443	439	eqeq2d	$⊢ t = x \to (i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow i = ⦋ 1^{st} (x) / s ⦌ C)$
444	443	rexbidv	$⊢ t = x \to (\exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)$
445	444	ralbidv	$⊢ t = x \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)$
446	442 445	anbi12d	$⊢ t = x \to ((N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C) \leftrightarrow (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C))$
447	446	rexrab	$⊢ \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = s \leftrightarrow \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s)$
448		xp1st	$⊢ x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 1^{st} (x) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})$
449	448	anim1i	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \to (1^{st} (x) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)$
450		eleq1	$⊢ 1^{st} (x) = s \to (1^{st} (x) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \leftrightarrow s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}))$
451		csbeq1a	$⊢ s = 1^{st} (x) \to C = ⦋ 1^{st} (x) / s ⦌ C$
452	451	eqcoms	$⊢ 1^{st} (x) = s \to C = ⦋ 1^{st} (x) / s ⦌ C$
453	452	eqcomd	$⊢ 1^{st} (x) = s \to ⦋ 1^{st} (x) / s ⦌ C = C$
454	453	eqeq2d	$⊢ 1^{st} (x) = s \to (i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow i = C)$
455	454	rexbidv	$⊢ 1^{st} (x) = s \to (\exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \exists j \in (0 \dots N) i = C)$
456	455	ralbidv	$⊢ 1^{st} (x) = s \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C)$
457	450 456	anbi12d	$⊢ 1^{st} (x) = s \to ((1^{st} (x) \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \leftrightarrow (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C))$
458	449 457	syl5ibcom	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \to (1^{st} (x) = s \to (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C))$
459	458	adantrl	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)) \to (1^{st} (x) = s \to (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C))$
460	459	expimpd	$⊢ x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to (((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s) \to (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C))$
461	460	rexlimiv	$⊢ \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s) \to (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C)$
462		simplr	$⊢ ((φ \land s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})$
463		ovex	$⊢ (0 \dots N) \in V$
464	463	enref	$⊢ (0 \dots N) \approx (0 \dots N)$
465		phpreu	$⊢ ((0 \dots N) \in Fin \land (0 \dots N) \approx (0 \dots N)) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C \leftrightarrow \forall i \in (0 \dots N) \exists! j \in (0 \dots N) i = C)$
466	23 464 465	mp2an	$⊢ \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C \leftrightarrow \forall i \in (0 \dots N) \exists! j \in (0 \dots N) i = C$
467	466	biimpi	$⊢ \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C \to \forall i \in (0 \dots N) \exists! j \in (0 \dots N) i = C$
468		eqeq1	$⊢ i = N \to (i = C \leftrightarrow N = C)$
469	468	reubidv	$⊢ i = N \to (\exists! j \in (0 \dots N) i = C \leftrightarrow \exists! j \in (0 \dots N) N = C)$
470	469	rspcva	$⊢ (N \in (0 \dots N) \land \forall i \in (0 \dots N) \exists! j \in (0 \dots N) i = C) \to \exists! j \in (0 \dots N) N = C$
471	232 467 470	syl2an	$⊢ (φ \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to \exists! j \in (0 \dots N) N = C$
472		riotacl	$⊢ \exists! j \in (0 \dots N) N = C \to (ι j \in (0 \dots N) \| N = C) \in (0 \dots N)$
473	471 472	syl	$⊢ (φ \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to (ι j \in (0 \dots N) \| N = C) \in (0 \dots N)$
474	473	adantlr	$⊢ ((φ \land s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to (ι j \in (0 \dots N) \| N = C) \in (0 \dots N)$
475		opelxpi	$⊢ (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land (ι j \in (0 \dots N) \| N = C) \in (0 \dots N)) \to ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
476	462 474 475	syl2anc	$⊢ ((φ \land s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
477		riotasbc	$⊢ \exists! j \in (0 \dots N) N = C \to \underset{˙}{[} (ι j \in (0 \dots N) \| N = C) / j \underset{˙}{]} N = C$
478	471 477	syl	$⊢ (φ \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to \underset{˙}{[} (ι j \in (0 \dots N) \| N = C) / j \underset{˙}{]} N = C$
479		riotaex	$⊢ (ι j \in (0 \dots N) \| N = C) \in V$
480		sbceq2g	$⊢ (ι j \in (0 \dots N) \| N = C) \in V \to (\underset{˙}{[} (ι j \in (0 \dots N) \| N = C) / j \underset{˙}{]} N = C \leftrightarrow N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C)$
481	479 480	ax-mp	$⊢ \underset{˙}{[} (ι j \in (0 \dots N) \| N = C) / j \underset{˙}{]} N = C \leftrightarrow N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C$
482	478 481	sylib	$⊢ (φ \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C$
483	482	expcom	$⊢ \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C \to (φ \to N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C)$
484	483	imdistanri	$⊢ (φ \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to (N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C)$
485	484	adantlr	$⊢ ((φ \land s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to (N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C)$
486		vex	$⊢ s \in V$
487	486 479	op2ndd	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to 2^{nd} (x) = (ι j \in (0 \dots N) \| N = C)$
488	487	csbeq1d	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to ⦋ 2^{nd} (x) / j ⦌ C = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C$
489		nfcv	$⊢ \underline{Ⅎ} j s$
490		nfriota1	$⊢ \underline{Ⅎ} j (ι j \in (0 \dots N) \| N = C)$
491	489 490	nfop	$⊢ \underline{Ⅎ} j ⟨s, (ι j \in (0 \dots N) \| N = C)⟩$
492	491	nfeq2	$⊢ Ⅎ j x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩$
493	486 479	op1std	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to 1^{st} (x) = s$
494	493	eqcomd	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to s = 1^{st} (x)$
495	494 451	syl	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to C = ⦋ 1^{st} (x) / s ⦌ C$
496	492 495	csbeq2d	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to ⦋ 2^{nd} (x) / j ⦌ C = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
497	488 496	eqtr3d	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
498	497	eqeq2d	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to (N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C \leftrightarrow N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C)$
499	495	eqeq2d	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to (i = C \leftrightarrow i = ⦋ 1^{st} (x) / s ⦌ C)$
500	492 499	rexbid	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to (\exists j \in (0 \dots N) i = C \leftrightarrow \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)$
501	500	ralbidv	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C \leftrightarrow \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)$
502	498 501	anbi12d	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to ((N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \leftrightarrow (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C))$
503	493	biantrud	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \leftrightarrow ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s))$
504	502 503	bitr2d	$⊢ x = ⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \to (((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s) \leftrightarrow (N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C))$
505	504	rspcev	$⊢ (⟨s, (ι j \in (0 \dots N) \| N = C)⟩ \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (N = ⦋ (ι j \in (0 \dots N) \| N = C) / j ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C)) \to \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s)$
506	476 485 505	syl2anc	$⊢ ((φ \land s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\})) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s)$
507	506	expl	$⊢ φ \to ((s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C) \to \exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s))$
508	461 507	impbid2	$⊢ φ \to (\exists x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \land 1^{st} (x) = s) \leftrightarrow (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C))$
509	447 508	bitrid	$⊢ φ \to (\exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = s \leftrightarrow (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C))$
510	509	abbidv	$⊢ φ \to \{s \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = s\} = \{s \| (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C)\}$
511		dfimafn	$⊢ (Fun 1^{st} \land \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \subseteq dom 1^{st}) \to 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}] = \{y \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = y\}$
512	428 433 511	mp2an	$⊢ 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}] = \{y \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = y\}$
513		nfcv	$⊢ \underline{Ⅎ} s 2^{nd} (t)$
514		nfcsb1v	$⊢ \underline{Ⅎ} s ⦋ 1^{st} (t) / s ⦌ C$
515	513 514	nfcsbw	$⊢ \underline{Ⅎ} s ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
516	515	nfeq2	$⊢ Ⅎ s N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
517		nfcv	$⊢ \underline{Ⅎ} s (0 \dots N)$
518	514	nfeq2	$⊢ Ⅎ s i = ⦋ 1^{st} (t) / s ⦌ C$
519	517 518	nfrexw	$⊢ Ⅎ s \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C$
520	517 519	nfralw	$⊢ Ⅎ s \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C$
521	516 520	nfan	$⊢ Ⅎ s (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)$
522		nfcv	$⊢ \underline{Ⅎ} s ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
523	521 522	nfrabw	$⊢ \underline{Ⅎ} s \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}$
524		nfv	$⊢ Ⅎ s 1^{st} (x) = y$
525	523 524	nfrexw	$⊢ Ⅎ s \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = y$
526		nfv	$⊢ Ⅎ y \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = s$
527		eqeq2	$⊢ y = s \to (1^{st} (x) = y \leftrightarrow 1^{st} (x) = s)$
528	527	rexbidv	$⊢ y = s \to (\exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = y \leftrightarrow \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = s)$
529	525 526 528	cbvabw	$⊢ \{y \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = y\} = \{s \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = s\}$
530	512 529	eqtri	$⊢ 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}] = \{s \| \exists x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} 1^{st} (x) = s\}$
531		df-rab	$⊢ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\} = \{s \| (s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C)\}$
532	510 530 531	3eqtr4g	$⊢ φ \to 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}] = \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
533		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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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) \exists j \in (0 \dots N) i = C\} \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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}] \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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \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) \exists j \in (0 \dots N) i = C\})$
534	532 533	syl	$⊢ φ \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{onto}{⟶} 1^{st} [\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}] \leftrightarrow ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\})$
535	435 534	mpbii	$⊢ φ \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
536		fof	$⊢ ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\} \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} ⟶ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
537	535 536	syl	$⊢ φ \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} ⟶ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
538		fvres	$⊢ x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (x) = 1^{st} (x)$
539		fvres	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (y) = 1^{st} (y)$
540	538 539	eqeqan12d	$⊢ (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}) \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (y) \leftrightarrow 1^{st} (x) = 1^{st} (y))$
541	540	adantl	$⊢ (φ \land (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\})) \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (y) \leftrightarrow 1^{st} (x) = 1^{st} (y))$
542	446	elrab	$⊢ x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \leftrightarrow (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C))$
543		xp2nd	$⊢ x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 2^{nd} (x) \in (0 \dots N)$
544	543	anim1i	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)) \to (2^{nd} (x) \in (0 \dots N) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C))$
545	542 544	sylbi	$⊢ x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \to (2^{nd} (x) \in (0 \dots N) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C))$
546		simpl	$⊢ (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C) \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
547	546	a1i	$⊢ t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to ((N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C) \to N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C)$
548	547	ss2rabi	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \subseteq \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C\}$
549	548	sseli	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \to y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C\}$
550		fveq2	$⊢ t = y \to 2^{nd} (t) = 2^{nd} (y)$
551	550	csbeq1d	$⊢ t = y \to ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C$
552		fveq2	$⊢ t = y \to 1^{st} (t) = 1^{st} (y)$
553	552	csbeq1d	$⊢ t = y \to ⦋ 1^{st} (t) / s ⦌ C = ⦋ 1^{st} (y) / s ⦌ C$
554	553	csbeq2dv	$⊢ t = y \to ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C$
555	551 554	eqtrd	$⊢ t = y \to ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C$
556	555	eqeq2d	$⊢ t = y \to (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \leftrightarrow N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)$
557	556	elrab	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C\} \leftrightarrow (y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)$
558		xp2nd	$⊢ y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \to 2^{nd} (y) \in (0 \dots N)$
559	558	anim1i	$⊢ (y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C) \to (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)$
560	557 559	sylbi	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C\} \to (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)$
561	549 560	syl	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \to (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)$
562	545 561	anim12i	$⊢ (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}) \to ((2^{nd} (x) \in (0 \dots N) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C))$
563		an4	$⊢ ((2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)) \leftrightarrow ((2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N)) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C))$
564	563	anbi2i	$⊢ (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land ((2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C))) \leftrightarrow (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land ((2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N)) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)))$
565		anass	$⊢ ((2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \leftrightarrow (2^{nd} (x) \in (0 \dots N) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C))$
566		ancom	$⊢ ((2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \leftrightarrow (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C))$
567	565 566	bitr3i	$⊢ (2^{nd} (x) \in (0 \dots N) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)) \leftrightarrow (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C))$
568	567	anbi1i	$⊢ ((2^{nd} (x) \in (0 \dots N) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)) \leftrightarrow ((\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C)) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C))$
569		anass	$⊢ ((\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C)) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)) \leftrightarrow (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land ((2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)))$
570	568 569	bitri	$⊢ ((2^{nd} (x) \in (0 \dots N) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)) \leftrightarrow (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land ((2^{nd} (x) \in (0 \dots N) \land N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)))$
571		anass	$⊢ ((\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N))) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)) \leftrightarrow (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land ((2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N)) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)))$
572	564 570 571	3bitr4i	$⊢ ((2^{nd} (x) \in (0 \dots N) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)) \land (2^{nd} (y) \in (0 \dots N) \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)) \leftrightarrow ((\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N))) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C))$
573	562 572	sylib	$⊢ (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}) \to ((\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N))) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C))$
574		phpreu	$⊢ ((0 \dots N) \in Fin \land (0 \dots N) \approx (0 \dots N)) \to (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N) \exists! j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C)$
575	23 464 574	mp2an	$⊢ \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \forall i \in (0 \dots N) \exists! j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C$
576		reurmo	$⊢ \exists! j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \to \exists^{*} j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C$
577	576	ralimi	$⊢ \forall i \in (0 \dots N) \exists! j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \to \forall i \in (0 \dots N) \exists^{*} j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C$
578	575 577	sylbi	$⊢ \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \to \forall i \in (0 \dots N) \exists^{*} j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C$
579		eqeq1	$⊢ i = N \to (i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow N = ⦋ 1^{st} (x) / s ⦌ C)$
580	579	rmobidv	$⊢ i = N \to (\exists^{} j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \exists^{} j \in (0 \dots N) N = ⦋ 1^{st} (x) / s ⦌ C)$
581	580	rspcva	$⊢ (N \in (0 \dots N) \land \forall i \in (0 \dots N) \exists^{} j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \to \exists^{} j \in (0 \dots N) N = ⦋ 1^{st} (x) / s ⦌ C$
582	232 578 581	syl2an	$⊢ (φ \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \to \exists^{*} j \in (0 \dots N) N = ⦋ 1^{st} (x) / s ⦌ C$
583		nfv	$⊢ Ⅎ k N = ⦋ 1^{st} (x) / s ⦌ C$
584	583	rmo3	$⊢ \exists^{*} j \in (0 \dots N) N = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \forall j \in (0 \dots N) \forall k \in (0 \dots N) ((N = ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \to j = k)$
585	582 584	sylib	$⊢ (φ \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \to \forall j \in (0 \dots N) \forall k \in (0 \dots N) ((N = ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \to j = k)$
586		nfcsb1v	$⊢ \underline{Ⅎ} j ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
587	586	nfeq2	$⊢ Ⅎ j N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
588		nfs1v	$⊢ Ⅎ j [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C$
589	587 588	nfan	$⊢ Ⅎ j (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C)$
590		nfv	$⊢ Ⅎ j 2^{nd} (x) = k$
591	589 590	nfim	$⊢ Ⅎ j ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = k)$
592		nfv	$⊢ Ⅎ k ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = 2^{nd} (y))$
593		csbeq1a	$⊢ j = 2^{nd} (x) \to ⦋ 1^{st} (x) / s ⦌ C = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
594	593	eqeq2d	$⊢ j = 2^{nd} (x) \to (N = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C)$
595	594	anbi1d	$⊢ j = 2^{nd} (x) \to ((N = ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \leftrightarrow (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C))$
596		eqeq1	$⊢ j = 2^{nd} (x) \to (j = k \leftrightarrow 2^{nd} (x) = k)$
597	595 596	imbi12d	$⊢ j = 2^{nd} (x) \to (((N = ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \to j = k) \leftrightarrow ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = k))$
598		sbsbc	$⊢ [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow \underset{˙}{[} k / j \underset{˙}{]} N = ⦋ 1^{st} (x) / s ⦌ C$
599		vex	$⊢ k \in V$
600		sbceq2g	$⊢ k \in V \to (\underset{˙}{[} k / j \underset{˙}{]} N = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow N = ⦋ k / j ⦌ ⦋ 1^{st} (x) / s ⦌ C)$
601	599 600	ax-mp	$⊢ \underset{˙}{[} k / j \underset{˙}{]} N = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow N = ⦋ k / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
602	598 601	bitri	$⊢ [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow N = ⦋ k / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
603		csbeq1	$⊢ k = 2^{nd} (y) \to ⦋ k / j ⦌ ⦋ 1^{st} (x) / s ⦌ C = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C$
604	603	eqeq2d	$⊢ k = 2^{nd} (y) \to (N = ⦋ k / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C)$
605	602 604	bitrid	$⊢ k = 2^{nd} (y) \to ([k/ j] N = ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C)$
606	605	anbi2d	$⊢ k = 2^{nd} (y) \to ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \leftrightarrow (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C))$
607		eqeq2	$⊢ k = 2^{nd} (y) \to (2^{nd} (x) = k \leftrightarrow 2^{nd} (x) = 2^{nd} (y))$
608	606 607	imbi12d	$⊢ k = 2^{nd} (y) \to (((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = k) \leftrightarrow ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = 2^{nd} (y)))$
609	591 592 597 608	rspc2	$⊢ (2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N)) \to (\forall j \in (0 \dots N) \forall k \in (0 \dots N) ((N = ⦋ 1^{st} (x) / s ⦌ C \land [k/ j] N = ⦋ 1^{st} (x) / s ⦌ C) \to j = k) \to ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = 2^{nd} (y)))$
610	585 609	syl5com	$⊢ (φ \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C) \to ((2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N)) \to ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = 2^{nd} (y)))$
611	610	impr	$⊢ (φ \land (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N)))) \to ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = 2^{nd} (y))$
612		csbeq1	$⊢ 1^{st} (x) = 1^{st} (y) \to ⦋ 1^{st} (x) / s ⦌ C = ⦋ 1^{st} (y) / s ⦌ C$
613	612	csbeq2dv	$⊢ 1^{st} (x) = 1^{st} (y) \to ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C$
614	613	eqeq2d	$⊢ 1^{st} (x) = 1^{st} (y) \to (N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \leftrightarrow N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C)$
615	614	anbi2d	$⊢ 1^{st} (x) = 1^{st} (y) \to ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \leftrightarrow (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C))$
616	615	imbi1d	$⊢ 1^{st} (x) = 1^{st} (y) \to (((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C) \to 2^{nd} (x) = 2^{nd} (y)) \leftrightarrow ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C) \to 2^{nd} (x) = 2^{nd} (y)))$
617	611 616	syl5ibcom	$⊢ (φ \land (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N)))) \to (1^{st} (x) = 1^{st} (y) \to ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C) \to 2^{nd} (x) = 2^{nd} (y)))$
618	617	com23	$⊢ (φ \land (\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N)))) \to ((N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C) \to (1^{st} (x) = 1^{st} (y) \to 2^{nd} (x) = 2^{nd} (y)))$
619	618	impr	$⊢ (φ \land ((\forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (x) / s ⦌ C \land (2^{nd} (x) \in (0 \dots N) \land 2^{nd} (y) \in (0 \dots N))) \land (N = ⦋ 2^{nd} (x) / j ⦌ ⦋ 1^{st} (x) / s ⦌ C \land N = ⦋ 2^{nd} (y) / j ⦌ ⦋ 1^{st} (y) / s ⦌ C))) \to (1^{st} (x) = 1^{st} (y) \to 2^{nd} (x) = 2^{nd} (y))$
620	573 619	sylan2	$⊢ (φ \land (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\})) \to (1^{st} (x) = 1^{st} (y) \to 2^{nd} (x) = 2^{nd} (y))$
621		elrabi	$⊢ x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \to x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
622		elrabi	$⊢ y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \to y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))$
623		xpopth	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((1^{st} (x) = 1^{st} (y) \land 2^{nd} (x) = 2^{nd} (y)) \leftrightarrow x = y)$
624	623	biimpd	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to ((1^{st} (x) = 1^{st} (y) \land 2^{nd} (x) = 2^{nd} (y)) \to x = y)$
625	624	expd	$⊢ (x \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \land y \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N))) \to (1^{st} (x) = 1^{st} (y) \to (2^{nd} (x) = 2^{nd} (y) \to x = y))$
626	621 622 625	syl2an	$⊢ (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}) \to (1^{st} (x) = 1^{st} (y) \to (2^{nd} (x) = 2^{nd} (y) \to x = y))$
627	626	adantl	$⊢ (φ \land (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\})) \to (1^{st} (x) = 1^{st} (y) \to (2^{nd} (x) = 2^{nd} (y) \to x = y))$
628	620 627	mpdd	$⊢ (φ \land (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\})) \to (1^{st} (x) = 1^{st} (y) \to x = y)$
629	541 628	sylbid	$⊢ (φ \land (x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \land y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\})) \to (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (y) \to x = y)$
630	629	ralrimivva	$⊢ φ \to \forall x \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \forall y \in \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} (({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (x) = ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (y) \to x = y)$
631		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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \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) \exists j \in (0 \dots N) i = C\} \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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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) \exists j \in (0 \dots N) i = C\} \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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} (({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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) (y) \to x = y))$
632	537 630 631	sylanbrc	$⊢ φ \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{1-1}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
633		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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \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) \exists j \in (0 \dots N) i = C\} \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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \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) \exists j \in (0 \dots N) i = C\} \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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) 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)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \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) \exists j \in (0 \dots N) i = C\})$
634	632 535 633	sylanbrc	$⊢ φ \to ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{1-1 onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
635		rabfi	$⊢ (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N) \in Fin \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \in Fin$
636	138 635	ax-mp	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \in Fin$
637	636	elexi	$⊢ \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \in V$
638	637	f1oen	$⊢ ({1^{st} ↾}_{\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}}) : \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \underset{1-1 onto}{⟶} \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\} \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \approx \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
639	634 638	syl	$⊢ φ \to \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \approx \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
640		rabfi	$⊢ ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\} \in Fin \to \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\} \in Fin$
641	136 640	ax-mp	$⊢ \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\} \in Fin$
642		hashen	$⊢ (\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \in Fin \land \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\} \in Fin) \to (\|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\| = \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}\| \leftrightarrow \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \approx \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\})$
643	636 641 642	mp2an	$⊢ \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\| = \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}\| \leftrightarrow \{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\} \approx \{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}$
644	639 643	sylibr	$⊢ φ \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\| = \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}\|$
645	644	oveq2d	$⊢ φ \to \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| (N = ⦋ 2^{nd} (t) / j ⦌ ⦋ 1^{st} (t) / s ⦌ C \land \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = ⦋ 1^{st} (t) / s ⦌ C)\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}\|$
646	202 425 645	3eqtr3d	$⊢ φ \to \sum_{x \in ({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}} \|\{x\} \times \{y \in (0 \dots N) \| (\forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{y\}) i = ⦋ x / s ⦌ C \land \forall j \in (0 \dots N) N \neq ⦋ x / s ⦌ C)\}\| = \|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}\|$
647	135 646	breqtrd	$⊢ φ \to 2 ∥ (\|\{t \in ((({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \times (0 \dots N)) \| \forall i \in (0 \dots N - 1) \exists j \in ((0 \dots N) ∖ \{2^{nd} (t)\}) i = ⦋ 1^{st} (t) / s ⦌ C\}\| - \|\{s \in (({(0 ..^K)}^{(1 \dots N)}) \times \{f \| f : (1 \dots N) \underset{1-1 onto}{⟶} (1 \dots N)\}) \| \forall i \in (0 \dots N) \exists j \in (0 \dots N) i = C\}\|)$

Metamath Proof Explorer

Theorem poimirlem26

Proof