poimirlem28

Description: Lemma for poimir , a variant of Sperner's lemma. (Contributed by Brendan Leahy, 21-Aug-2020)

	Ref	Expression
Hypotheses	poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	poimirlem28.1	⊢ ( 𝑝 = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → 𝐵 = 𝐶 )
	poimirlem28.2	⊢ ( ( 𝜑 ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) → 𝐵 ∈ ( 0 ... 𝑁 ) )
	poimirlem28.3	⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ) → 𝐵 < 𝑛 )
	poimirlem28.4	⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝐾 ) ) → 𝐵 ≠ ( 𝑛 − 1 ) )
	poimirlem28.5	⊢ ( 𝜑 → 𝐾 ∈ ℕ )
Assertion	poimirlem28	⊢ ( 𝜑 → ∃ 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		poimirlem28.1	⊢ ( 𝑝 = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → 𝐵 = 𝐶 )
3		poimirlem28.2	⊢ ( ( 𝜑 ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) → 𝐵 ∈ ( 0 ... 𝑁 ) )
4		poimirlem28.3	⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ) → 𝐵 < 𝑛 )
5		poimirlem28.4	⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝐾 ) ) → 𝐵 ≠ ( 𝑛 − 1 ) )
6		poimirlem28.5	⊢ ( 𝜑 → 𝐾 ∈ ℕ )
7	1	nnnn0d	⊢ ( 𝜑 → 𝑁 ∈ ℕ₀ )
8	1	nnred	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
9	8	leidd	⊢ ( 𝜑 → 𝑁 ≤ 𝑁 )
10	7 7 9	3jca	⊢ ( 𝜑 → ( 𝑁 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑁 ≤ 𝑁 ) )
11		oveq2	⊢ ( 𝑘 = 0 → ( 1 ... 𝑘 ) = ( 1 ... 0 ) )
12		fz10	⊢ ( 1 ... 0 ) = ∅
13	11 12	eqtrdi	⊢ ( 𝑘 = 0 → ( 1 ... 𝑘 ) = ∅ )
14	13	oveq2d	⊢ ( 𝑘 = 0 → ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) = ( ( 0 ..^ 𝐾 ) ↑_m ∅ ) )
15		fzofi	⊢ ( 0 ..^ 𝐾 ) ∈ Fin
16		map0e	⊢ ( ( 0 ..^ 𝐾 ) ∈ Fin → ( ( 0 ..^ 𝐾 ) ↑_m ∅ ) = 1_o )
17	15 16	ax-mp	⊢ ( ( 0 ..^ 𝐾 ) ↑_m ∅ ) = 1_o
18		df1o2	⊢ 1_o = { ∅ }
19	17 18	eqtri	⊢ ( ( 0 ..^ 𝐾 ) ↑_m ∅ ) = { ∅ }
20	14 19	eqtrdi	⊢ ( 𝑘 = 0 → ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) = { ∅ } )
21		eqidd	⊢ ( 𝑘 = 0 → 𝑓 = 𝑓 )
22	21 13 13	f1oeq123d	⊢ ( 𝑘 = 0 → ( 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) ↔ 𝑓 : ∅ –1-1-onto→ ∅ ) )
23		eqid	⊢ ∅ = ∅
24		f1o00	⊢ ( 𝑓 : ∅ –1-1-onto→ ∅ ↔ ( 𝑓 = ∅ ∧ ∅ = ∅ ) )
25	23 24	mpbiran2	⊢ ( 𝑓 : ∅ –1-1-onto→ ∅ ↔ 𝑓 = ∅ )
26	22 25	bitrdi	⊢ ( 𝑘 = 0 → ( 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) ↔ 𝑓 = ∅ ) )
27	26	abbidv	⊢ ( 𝑘 = 0 → { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } = { 𝑓 ∣ 𝑓 = ∅ } )
28		df-sn	⊢ { ∅ } = { 𝑓 ∣ 𝑓 = ∅ }
29	27 28	eqtr4di	⊢ ( 𝑘 = 0 → { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } = { ∅ } )
30	20 29	xpeq12d	⊢ ( 𝑘 = 0 → ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) = ( { ∅ } × { ∅ } ) )
31		0ex	⊢ ∅ ∈ V
32	31 31	xpsn	⊢ ( { ∅ } × { ∅ } ) = { ⟨ ∅ , ∅ ⟩ }
33	30 32	eqtr2di	⊢ ( 𝑘 = 0 → { ⟨ ∅ , ∅ ⟩ } = ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) )
34		elsni	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → 𝑠 = ⟨ ∅ , ∅ ⟩ )
35	31 31	op1std	⊢ ( 𝑠 = ⟨ ∅ , ∅ ⟩ → ( 1^st ‘ 𝑠 ) = ∅ )
36	34 35	syl	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ( 1^st ‘ 𝑠 ) = ∅ )
37	36	oveq1d	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) = ( ∅ ∘_f + ∅ ) )
38		f0	⊢ ∅ : ∅ ⟶ ∅
39		ffn	⊢ ( ∅ : ∅ ⟶ ∅ → ∅ Fn ∅ )
40	38 39	mp1i	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ∅ Fn ∅ )
41	31	a1i	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ∅ ∈ V )
42		inidm	⊢ ( ∅ ∩ ∅ ) = ∅
43		0fv	⊢ ( ∅ ‘ 𝑛 ) = ∅
44	43	a1i	⊢ ( ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∧ 𝑛 ∈ ∅ ) → ( ∅ ‘ 𝑛 ) = ∅ )
45	40 40 41 41 42 44 44	offval	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ( ∅ ∘_f + ∅ ) = ( 𝑛 ∈ ∅ ↦ ( ∅ + ∅ ) ) )
46		mpt0	⊢ ( 𝑛 ∈ ∅ ↦ ( ∅ + ∅ ) ) = ∅
47	45 46	eqtrdi	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ( ∅ ∘_f + ∅ ) = ∅ )
48	37 47	eqtrd	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) = ∅ )
49	48	uneq1d	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) = ( ∅ ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) )
50		uncom	⊢ ( ∅ ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) = ( ( ( 1 ... 𝑁 ) × { 0 } ) ∪ ∅ )
51		un0	⊢ ( ( ( 1 ... 𝑁 ) × { 0 } ) ∪ ∅ ) = ( ( 1 ... 𝑁 ) × { 0 } )
52	50 51	eqtri	⊢ ( ∅ ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) = ( ( 1 ... 𝑁 ) × { 0 } )
53	49 52	eqtr2di	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ( ( 1 ... 𝑁 ) × { 0 } ) = ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) )
54	53	csbeq1d	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
55	54	eqeq2d	⊢ ( 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } → ( 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 ↔ 0 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
56		oveq2	⊢ ( 𝑘 = 0 → ( 0 ... 𝑘 ) = ( 0 ... 0 ) )
57		0z	⊢ 0 ∈ ℤ
58		fzsn	⊢ ( 0 ∈ ℤ → ( 0 ... 0 ) = { 0 } )
59	57 58	ax-mp	⊢ ( 0 ... 0 ) = { 0 }
60	56 59	eqtrdi	⊢ ( 𝑘 = 0 → ( 0 ... 𝑘 ) = { 0 } )
61		oveq2	⊢ ( 𝑘 = 0 → ( ( 𝑗 + 1 ) ... 𝑘 ) = ( ( 𝑗 + 1 ) ... 0 ) )
62	61	imaeq2d	⊢ ( 𝑘 = 0 → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) = ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) )
63	62	xpeq1d	⊢ ( 𝑘 = 0 → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) = ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) )
64	63	uneq2d	⊢ ( 𝑘 = 0 → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) )
65	64	oveq2d	⊢ ( 𝑘 = 0 → ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) )
66		oveq1	⊢ ( 𝑘 = 0 → ( 𝑘 + 1 ) = ( 0 + 1 ) )
67		0p1e1	⊢ ( 0 + 1 ) = 1
68	66 67	eqtrdi	⊢ ( 𝑘 = 0 → ( 𝑘 + 1 ) = 1 )
69	68	oveq1d	⊢ ( 𝑘 = 0 → ( ( 𝑘 + 1 ) ... 𝑁 ) = ( 1 ... 𝑁 ) )
70	69	xpeq1d	⊢ ( 𝑘 = 0 → ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) = ( ( 1 ... 𝑁 ) × { 0 } ) )
71	65 70	uneq12d	⊢ ( 𝑘 = 0 → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) )
72	71	csbeq1d	⊢ ( 𝑘 = 0 → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
73	72	eqeq2d	⊢ ( 𝑘 = 0 → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
74	60 73	rexeqbidv	⊢ ( 𝑘 = 0 → ( ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ { 0 } 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
75		c0ex	⊢ 0 ∈ V
76		oveq2	⊢ ( 𝑗 = 0 → ( 1 ... 𝑗 ) = ( 1 ... 0 ) )
77	76 12	eqtrdi	⊢ ( 𝑗 = 0 → ( 1 ... 𝑗 ) = ∅ )
78	77	imaeq2d	⊢ ( 𝑗 = 0 → ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ 𝑠 ) “ ∅ ) )
79		ima0	⊢ ( ( 2^nd ‘ 𝑠 ) “ ∅ ) = ∅
80	78 79	eqtrdi	⊢ ( 𝑗 = 0 → ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) = ∅ )
81	80	xpeq1d	⊢ ( 𝑗 = 0 → ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ∅ × { 1 } ) )
82		0xp	⊢ ( ∅ × { 1 } ) = ∅
83	81 82	eqtrdi	⊢ ( 𝑗 = 0 → ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ∅ )
84		oveq1	⊢ ( 𝑗 = 0 → ( 𝑗 + 1 ) = ( 0 + 1 ) )
85	84 67	eqtrdi	⊢ ( 𝑗 = 0 → ( 𝑗 + 1 ) = 1 )
86	85	oveq1d	⊢ ( 𝑗 = 0 → ( ( 𝑗 + 1 ) ... 0 ) = ( 1 ... 0 ) )
87	86 12	eqtrdi	⊢ ( 𝑗 = 0 → ( ( 𝑗 + 1 ) ... 0 ) = ∅ )
88	87	imaeq2d	⊢ ( 𝑗 = 0 → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) = ( ( 2^nd ‘ 𝑠 ) “ ∅ ) )
89	88 79	eqtrdi	⊢ ( 𝑗 = 0 → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) = ∅ )
90	89	xpeq1d	⊢ ( 𝑗 = 0 → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) = ( ∅ × { 0 } ) )
91		0xp	⊢ ( ∅ × { 0 } ) = ∅
92	90 91	eqtrdi	⊢ ( 𝑗 = 0 → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) = ∅ )
93	83 92	uneq12d	⊢ ( 𝑗 = 0 → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) = ( ∅ ∪ ∅ ) )
94		un0	⊢ ( ∅ ∪ ∅ ) = ∅
95	93 94	eqtrdi	⊢ ( 𝑗 = 0 → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) = ∅ )
96	95	oveq2d	⊢ ( 𝑗 = 0 → ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) )
97	96	uneq1d	⊢ ( 𝑗 = 0 → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) )
98	97	csbeq1d	⊢ ( 𝑗 = 0 → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
99	98	eqeq2d	⊢ ( 𝑗 = 0 → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
100	75 99	rexsn	⊢ ( ∃ 𝑗 ∈ { 0 } 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 0 ) ) × { 0 } ) ) ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
101	74 100	bitrdi	⊢ ( 𝑘 = 0 → ( ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
102	60 101	raleqbidv	⊢ ( 𝑘 = 0 → ( ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ { 0 } 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
103		eqeq1	⊢ ( 𝑖 = 0 → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 0 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
104	75 103	ralsn	⊢ ( ∀ 𝑖 ∈ { 0 } 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 0 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
105	102 104	bitr2di	⊢ ( 𝑘 = 0 → ( 0 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ∅ ) ∪ ( ( 1 ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
106	55 105	sylan9bbr	⊢ ( ( 𝑘 = 0 ∧ 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ) → ( 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
107	33 106	rabeqbidva	⊢ ( 𝑘 = 0 → { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } = { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } )
108	107	eqcomd	⊢ ( 𝑘 = 0 → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } = { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } )
109	108	fveq2d	⊢ ( 𝑘 = 0 → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) = ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) )
110	109	breq2d	⊢ ( 𝑘 = 0 → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) ) )
111	110	notbid	⊢ ( 𝑘 = 0 → ( ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) ) )
112	111	imbi2d	⊢ ( 𝑘 = 0 → ( ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ↔ ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) ) ) )
113		oveq2	⊢ ( 𝑘 = 𝑚 → ( 1 ... 𝑘 ) = ( 1 ... 𝑚 ) )
114	113	oveq2d	⊢ ( 𝑘 = 𝑚 → ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) = ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) )
115		eqidd	⊢ ( 𝑘 = 𝑚 → 𝑓 = 𝑓 )
116	115 113 113	f1oeq123d	⊢ ( 𝑘 = 𝑚 → ( 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) ↔ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) ) )
117	116	abbidv	⊢ ( 𝑘 = 𝑚 → { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } = { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } )
118	114 117	xpeq12d	⊢ ( 𝑘 = 𝑚 → ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) = ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) )
119		oveq2	⊢ ( 𝑘 = 𝑚 → ( 0 ... 𝑘 ) = ( 0 ... 𝑚 ) )
120		oveq2	⊢ ( 𝑘 = 𝑚 → ( ( 𝑗 + 1 ) ... 𝑘 ) = ( ( 𝑗 + 1 ) ... 𝑚 ) )
121	120	imaeq2d	⊢ ( 𝑘 = 𝑚 → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) = ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) )
122	121	xpeq1d	⊢ ( 𝑘 = 𝑚 → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) = ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) )
123	122	uneq2d	⊢ ( 𝑘 = 𝑚 → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) )
124	123	oveq2d	⊢ ( 𝑘 = 𝑚 → ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) )
125		oveq1	⊢ ( 𝑘 = 𝑚 → ( 𝑘 + 1 ) = ( 𝑚 + 1 ) )
126	125	oveq1d	⊢ ( 𝑘 = 𝑚 → ( ( 𝑘 + 1 ) ... 𝑁 ) = ( ( 𝑚 + 1 ) ... 𝑁 ) )
127	126	xpeq1d	⊢ ( 𝑘 = 𝑚 → ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) = ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) )
128	124 127	uneq12d	⊢ ( 𝑘 = 𝑚 → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) )
129	128	csbeq1d	⊢ ( 𝑘 = 𝑚 → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
130	129	eqeq2d	⊢ ( 𝑘 = 𝑚 → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
131	119 130	rexeqbidv	⊢ ( 𝑘 = 𝑚 → ( ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
132	119 131	raleqbidv	⊢ ( 𝑘 = 𝑚 → ( ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
133	118 132	rabeqbidv	⊢ ( 𝑘 = 𝑚 → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } = { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } )
134	133	fveq2d	⊢ ( 𝑘 = 𝑚 → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) )
135	134	breq2d	⊢ ( 𝑘 = 𝑚 → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
136	135	notbid	⊢ ( 𝑘 = 𝑚 → ( ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
137	136	imbi2d	⊢ ( 𝑘 = 𝑚 → ( ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ↔ ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) )
138		oveq2	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( 1 ... 𝑘 ) = ( 1 ... ( 𝑚 + 1 ) ) )
139	138	oveq2d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) = ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) )
140		eqidd	⊢ ( 𝑘 = ( 𝑚 + 1 ) → 𝑓 = 𝑓 )
141	140 138 138	f1oeq123d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) ↔ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) ) )
142	141	abbidv	⊢ ( 𝑘 = ( 𝑚 + 1 ) → { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } = { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } )
143	139 142	xpeq12d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) = ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) )
144		oveq2	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( 0 ... 𝑘 ) = ( 0 ... ( 𝑚 + 1 ) ) )
145		oveq2	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( 𝑗 + 1 ) ... 𝑘 ) = ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) )
146	145	imaeq2d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) = ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) )
147	146	xpeq1d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) = ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) )
148	147	uneq2d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) )
149	148	oveq2d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) )
150		oveq1	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( 𝑘 + 1 ) = ( ( 𝑚 + 1 ) + 1 ) )
151	150	oveq1d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( 𝑘 + 1 ) ... 𝑁 ) = ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) )
152	151	xpeq1d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) = ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) )
153	149 152	uneq12d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
154	153	csbeq1d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
155	154	eqeq2d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
156	144 155	rexeqbidv	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
157	144 156	raleqbidv	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
158	143 157	rabeqbidv	⊢ ( 𝑘 = ( 𝑚 + 1 ) → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } = { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } )
159	158	fveq2d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) )
160	159	breq2d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
161	160	notbid	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
162	161	imbi2d	⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ↔ ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) )
163		oveq2	⊢ ( 𝑘 = 𝑁 → ( 1 ... 𝑘 ) = ( 1 ... 𝑁 ) )
164	163	oveq2d	⊢ ( 𝑘 = 𝑁 → ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) = ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
165		eqidd	⊢ ( 𝑘 = 𝑁 → 𝑓 = 𝑓 )
166	165 163 163	f1oeq123d	⊢ ( 𝑘 = 𝑁 → ( 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) ↔ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) )
167	166	abbidv	⊢ ( 𝑘 = 𝑁 → { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } = { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
168	164 167	xpeq12d	⊢ ( 𝑘 = 𝑁 → ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) = ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) )
169		oveq2	⊢ ( 𝑘 = 𝑁 → ( 0 ... 𝑘 ) = ( 0 ... 𝑁 ) )
170		oveq2	⊢ ( 𝑘 = 𝑁 → ( ( 𝑗 + 1 ) ... 𝑘 ) = ( ( 𝑗 + 1 ) ... 𝑁 ) )
171	170	imaeq2d	⊢ ( 𝑘 = 𝑁 → ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) = ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) )
172	171	xpeq1d	⊢ ( 𝑘 = 𝑁 → ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) = ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) )
173	172	uneq2d	⊢ ( 𝑘 = 𝑁 → ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
174	173	oveq2d	⊢ ( 𝑘 = 𝑁 → ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
175		oveq1	⊢ ( 𝑘 = 𝑁 → ( 𝑘 + 1 ) = ( 𝑁 + 1 ) )
176	175	oveq1d	⊢ ( 𝑘 = 𝑁 → ( ( 𝑘 + 1 ) ... 𝑁 ) = ( ( 𝑁 + 1 ) ... 𝑁 ) )
177	176	xpeq1d	⊢ ( 𝑘 = 𝑁 → ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) = ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) )
178	174 177	uneq12d	⊢ ( 𝑘 = 𝑁 → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) )
179	178	csbeq1d	⊢ ( 𝑘 = 𝑁 → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
180	179	eqeq2d	⊢ ( 𝑘 = 𝑁 → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
181	169 180	rexeqbidv	⊢ ( 𝑘 = 𝑁 → ( ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
182	169 181	raleqbidv	⊢ ( 𝑘 = 𝑁 → ( ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
183	168 182	rabeqbidv	⊢ ( 𝑘 = 𝑁 → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } = { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } )
184	183	fveq2d	⊢ ( 𝑘 = 𝑁 → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) )
185	184	breq2d	⊢ ( 𝑘 = 𝑁 → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
186	185	notbid	⊢ ( 𝑘 = 𝑁 → ( ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
187	186	imbi2d	⊢ ( 𝑘 = 𝑁 → ( ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑘 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑘 ) –1-1-onto→ ( 1 ... 𝑘 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑘 ) ∃ 𝑗 ∈ ( 0 ... 𝑘 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑘 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑘 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ↔ ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) )
188		n2dvds1	⊢ ¬ 2 ∥ 1
189		opex	⊢ ⟨ ∅ , ∅ ⟩ ∈ V
190		hashsng	⊢ ( ⟨ ∅ , ∅ ⟩ ∈ V → ( ♯ ‘ { ⟨ ∅ , ∅ ⟩ } ) = 1 )
191	189 190	ax-mp	⊢ ( ♯ ‘ { ⟨ ∅ , ∅ ⟩ } ) = 1
192		nnuz	⊢ ℕ = ( ℤ_≥ ‘ 1 )
193	1 192	eleqtrdi	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ_≥ ‘ 1 ) )
194		eluzfz1	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑁 ) )
195	193 194	syl	⊢ ( 𝜑 → 1 ∈ ( 1 ... 𝑁 ) )
196	6	nnnn0d	⊢ ( 𝜑 → 𝐾 ∈ ℕ₀ )
197		0elfz	⊢ ( 𝐾 ∈ ℕ₀ → 0 ∈ ( 0 ... 𝐾 ) )
198		fconst6g	⊢ ( 0 ∈ ( 0 ... 𝐾 ) → ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
199	196 197 198	3syl	⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
200	75	fvconst2	⊢ ( 1 ∈ ( 1 ... 𝑁 ) → ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 )
201	195 200	syl	⊢ ( 𝜑 → ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 )
202	195 199 201	3jca	⊢ ( 𝜑 → ( 1 ∈ ( 1 ... 𝑁 ) ∧ ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 ) )
203		nfv	⊢ Ⅎ 𝑝 ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 ) )
204		nfcsb1v	⊢ Ⅎ 𝑝 ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵
205	204	nfeq1	⊢ Ⅎ 𝑝 ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 = 0
206	203 205	nfim	⊢ Ⅎ 𝑝 ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 ) ) → ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 = 0 )
207		ovex	⊢ ( 1 ... 𝑁 ) ∈ V
208		snex	⊢ { 0 } ∈ V
209	207 208	xpex	⊢ ( ( 1 ... 𝑁 ) × { 0 } ) ∈ V
210		feq1	⊢ ( 𝑝 = ( ( 1 ... 𝑁 ) × { 0 } ) → ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ↔ ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) )
211		fveq1	⊢ ( 𝑝 = ( ( 1 ... 𝑁 ) × { 0 } ) → ( 𝑝 ‘ 1 ) = ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) )
212	211	eqeq1d	⊢ ( 𝑝 = ( ( 1 ... 𝑁 ) × { 0 } ) → ( ( 𝑝 ‘ 1 ) = 0 ↔ ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 ) )
213	210 212	3anbi23d	⊢ ( 𝑝 = ( ( 1 ... 𝑁 ) × { 0 } ) → ( ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ↔ ( 1 ∈ ( 1 ... 𝑁 ) ∧ ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 ) ) )
214	213	anbi2d	⊢ ( 𝑝 = ( ( 1 ... 𝑁 ) × { 0 } ) → ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ) ↔ ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 ) ) ) )
215		csbeq1a	⊢ ( 𝑝 = ( ( 1 ... 𝑁 ) × { 0 } ) → 𝐵 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 )
216	215	eqeq1d	⊢ ( 𝑝 = ( ( 1 ... 𝑁 ) × { 0 } ) → ( 𝐵 = 0 ↔ ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 = 0 ) )
217	214 216	imbi12d	⊢ ( 𝑝 = ( ( 1 ... 𝑁 ) × { 0 } ) → ( ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ) → 𝐵 = 0 ) ↔ ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 ) ) → ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 = 0 ) ) )
218		1ex	⊢ 1 ∈ V
219		eleq1	⊢ ( 𝑛 = 1 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↔ 1 ∈ ( 1 ... 𝑁 ) ) )
220		fveqeq2	⊢ ( 𝑛 = 1 → ( ( 𝑝 ‘ 𝑛 ) = 0 ↔ ( 𝑝 ‘ 1 ) = 0 ) )
221	219 220	3anbi13d	⊢ ( 𝑛 = 1 → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ↔ ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ) )
222	221	anbi2d	⊢ ( 𝑛 = 1 → ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ) ↔ ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ) ) )
223		breq2	⊢ ( 𝑛 = 1 → ( 𝐵 < 𝑛 ↔ 𝐵 < 1 ) )
224	222 223	imbi12d	⊢ ( 𝑛 = 1 → ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ) → 𝐵 < 𝑛 ) ↔ ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ) → 𝐵 < 1 ) ) )
225	218 224 4	vtocl	⊢ ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ) → 𝐵 < 1 )
226		elfznn0	⊢ ( 𝐵 ∈ ( 0 ... 𝑁 ) → 𝐵 ∈ ℕ₀ )
227		nn0lt10b	⊢ ( 𝐵 ∈ ℕ₀ → ( 𝐵 < 1 ↔ 𝐵 = 0 ) )
228	3 226 227	3syl	⊢ ( ( 𝜑 ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) → ( 𝐵 < 1 ↔ 𝐵 = 0 ) )
229	228	3ad2antr2	⊢ ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ) → ( 𝐵 < 1 ↔ 𝐵 = 0 ) )
230	225 229	mpbid	⊢ ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 1 ) = 0 ) ) → 𝐵 = 0 )
231	206 209 217 230	vtoclf	⊢ ( ( 𝜑 ∧ ( 1 ∈ ( 1 ... 𝑁 ) ∧ ( ( 1 ... 𝑁 ) × { 0 } ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( 1 ... 𝑁 ) × { 0 } ) ‘ 1 ) = 0 ) ) → ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 = 0 )
232	202 231	mpdan	⊢ ( 𝜑 → ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 = 0 )
233	232	eqcomd	⊢ ( 𝜑 → 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 )
234	233	ralrimivw	⊢ ( 𝜑 → ∀ 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 )
235		rabid2	⊢ ( { ⟨ ∅ , ∅ ⟩ } = { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ↔ ∀ 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 )
236	234 235	sylibr	⊢ ( 𝜑 → { ⟨ ∅ , ∅ ⟩ } = { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } )
237	236	fveq2d	⊢ ( 𝜑 → ( ♯ ‘ { ⟨ ∅ , ∅ ⟩ } ) = ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) )
238	191 237	eqtr3id	⊢ ( 𝜑 → 1 = ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) )
239	238	breq2d	⊢ ( 𝜑 → ( 2 ∥ 1 ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) ) )
240	188 239	mtbii	⊢ ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) )
241	240	a1i	⊢ ( 𝑁 ∈ ℕ₀ → ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ { ⟨ ∅ , ∅ ⟩ } ∣ 0 = ⦋ ( ( 1 ... 𝑁 ) × { 0 } ) / 𝑝 ⦌ 𝐵 } ) ) )
242		2z	⊢ 2 ∈ ℤ
243		fzfi	⊢ ( 1 ... ( 𝑚 + 1 ) ) ∈ Fin
244		mapfi	⊢ ( ( ( 0 ..^ 𝐾 ) ∈ Fin ∧ ( 1 ... ( 𝑚 + 1 ) ) ∈ Fin ) → ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) ∈ Fin )
245	15 243 244	mp2an	⊢ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) ∈ Fin
246		ovex	⊢ ( 1 ... ( 𝑚 + 1 ) ) ∈ V
247	246 246	mapval	⊢ ( ( 1 ... ( 𝑚 + 1 ) ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) = { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 1 ... ( 𝑚 + 1 ) ) }
248		mapfi	⊢ ( ( ( 1 ... ( 𝑚 + 1 ) ) ∈ Fin ∧ ( 1 ... ( 𝑚 + 1 ) ) ∈ Fin ) → ( ( 1 ... ( 𝑚 + 1 ) ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) ∈ Fin )
249	243 243 248	mp2an	⊢ ( ( 1 ... ( 𝑚 + 1 ) ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) ∈ Fin
250	247 249	eqeltrri	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 1 ... ( 𝑚 + 1 ) ) } ∈ Fin
251		f1of	⊢ ( 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) → 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 1 ... ( 𝑚 + 1 ) ) )
252	251	ss2abi	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ⊆ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 1 ... ( 𝑚 + 1 ) ) }
253		ssfi	⊢ ( ( { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 1 ... ( 𝑚 + 1 ) ) } ∈ Fin ∧ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ⊆ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 1 ... ( 𝑚 + 1 ) ) } ) → { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ∈ Fin )
254	250 252 253	mp2an	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ∈ Fin
255		xpfi	⊢ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) ∈ Fin ∧ { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ∈ Fin ) → ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∈ Fin )
256	245 254 255	mp2an	⊢ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∈ Fin
257		rabfi	⊢ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∈ Fin → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∈ Fin )
258		hashcl	⊢ ( { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∈ Fin → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℕ₀ )
259	256 257 258	mp2b	⊢ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℕ₀
260	259	nn0zi	⊢ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℤ
261		rabfi	⊢ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∈ Fin → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ∈ Fin )
262		hashcl	⊢ ( { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ∈ Fin → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ∈ ℕ₀ )
263	256 261 262	mp2b	⊢ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ∈ ℕ₀
264	263	nn0zi	⊢ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ∈ ℤ
265	242 260 264	3pm3.2i	⊢ ( 2 ∈ ℤ ∧ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℤ ∧ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ∈ ℤ )
266		nn0p1nn	⊢ ( 𝑚 ∈ ℕ₀ → ( 𝑚 + 1 ) ∈ ℕ )
267	266	ad2antrl	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 𝑚 + 1 ) ∈ ℕ )
268		uneq1	⊢ ( 𝑞 = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) )
269	268	csbeq1d	⊢ ( 𝑞 = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
270	75	fconst	⊢ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) : ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ⟶ { 0 }
271	270	jctr	⊢ ( 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) → ( 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) : ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ⟶ { 0 } ) )
272	266	nnred	⊢ ( 𝑚 ∈ ℕ₀ → ( 𝑚 + 1 ) ∈ ℝ )
273	272	ltp1d	⊢ ( 𝑚 ∈ ℕ₀ → ( 𝑚 + 1 ) < ( ( 𝑚 + 1 ) + 1 ) )
274		fzdisj	⊢ ( ( 𝑚 + 1 ) < ( ( 𝑚 + 1 ) + 1 ) → ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
275	273 274	syl	⊢ ( 𝑚 ∈ ℕ₀ → ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
276		fun	⊢ ( ( ( 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) : ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ⟶ { 0 } ) ∧ ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ ( ( 0 ... 𝐾 ) ∪ { 0 } ) )
277	271 275 276	syl2anr	⊢ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ ( ( 0 ... 𝐾 ) ∪ { 0 } ) )
278	277	adantlr	⊢ ( ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ ( ( 0 ... 𝐾 ) ∪ { 0 } ) )
279	278	adantl	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ ( ( 0 ... 𝐾 ) ∪ { 0 } ) )
280	266	peano2nnd	⊢ ( 𝑚 ∈ ℕ₀ → ( ( 𝑚 + 1 ) + 1 ) ∈ ℕ )
281	280 192	eleqtrdi	⊢ ( 𝑚 ∈ ℕ₀ → ( ( 𝑚 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
282	281	ad2antrl	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ( 𝑚 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
283		nn0z	⊢ ( 𝑚 ∈ ℕ₀ → 𝑚 ∈ ℤ )
284	1	nnzd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
285		zltp1le	⊢ ( ( 𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑚 < 𝑁 ↔ ( 𝑚 + 1 ) ≤ 𝑁 ) )
286	283 284 285	syl2anr	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) → ( 𝑚 < 𝑁 ↔ ( 𝑚 + 1 ) ≤ 𝑁 ) )
287	286	biimpa	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑚 < 𝑁 ) → ( 𝑚 + 1 ) ≤ 𝑁 )
288	287	anasss	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 𝑚 + 1 ) ≤ 𝑁 )
289	283	peano2zd	⊢ ( 𝑚 ∈ ℕ₀ → ( 𝑚 + 1 ) ∈ ℤ )
290	289	adantr	⊢ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) → ( 𝑚 + 1 ) ∈ ℤ )
291		eluz	⊢ ( ( ( 𝑚 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑚 + 1 ) ) ↔ ( 𝑚 + 1 ) ≤ 𝑁 ) )
292	290 284 291	syl2anr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑚 + 1 ) ) ↔ ( 𝑚 + 1 ) ≤ 𝑁 ) )
293	288 292	mpbird	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑚 + 1 ) ) )
294		fzsplit2	⊢ ( ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑚 + 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) )
295	282 293 294	syl2anc	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) )
296	295	eqcomd	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
297	196 197	syl	⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝐾 ) )
298	297	snssd	⊢ ( 𝜑 → { 0 } ⊆ ( 0 ... 𝐾 ) )
299		ssequn2	⊢ ( { 0 } ⊆ ( 0 ... 𝐾 ) ↔ ( ( 0 ... 𝐾 ) ∪ { 0 } ) = ( 0 ... 𝐾 ) )
300	298 299	sylib	⊢ ( 𝜑 → ( ( 0 ... 𝐾 ) ∪ { 0 } ) = ( 0 ... 𝐾 ) )
301	300	adantr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ( 0 ... 𝐾 ) ∪ { 0 } ) = ( 0 ... 𝐾 ) )
302	296 301	feq23d	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ ( ( 0 ... 𝐾 ) ∪ { 0 } ) ↔ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) )
303	302	adantrr	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( ( 1 ... ( 𝑚 + 1 ) ) ∪ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ ( ( 0 ... 𝐾 ) ∪ { 0 } ) ↔ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) )
304	279 303	mpbid	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
305		nfv	⊢ Ⅎ 𝑝 ( 𝜑 ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
306		nfcsb1v	⊢ Ⅎ 𝑝 ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵
307	306	nfel1	⊢ Ⅎ 𝑝 ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 )
308	305 307	nfim	⊢ Ⅎ 𝑝 ( ( 𝜑 ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) )
309		vex	⊢ 𝑞 ∈ V
310		ovex	⊢ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ∈ V
311	310 208	xpex	⊢ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ∈ V
312	309 311	unex	⊢ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ∈ V
313		feq1	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ↔ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) )
314	313	anbi2d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝜑 ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) ↔ ( 𝜑 ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) ) )
315		csbeq1a	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → 𝐵 = ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 )
316	315	eleq1d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( 𝐵 ∈ ( 0 ... 𝑁 ) ↔ ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) ) )
317	314 316	imbi12d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( ( 𝜑 ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) → 𝐵 ∈ ( 0 ... 𝑁 ) ) ↔ ( ( 𝜑 ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) ) ) )
318	308 312 317 3	vtoclf	⊢ ( ( 𝜑 ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) )
319	304 318	syldan	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) )
320	319	anassrs	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) )
321		elfznn0	⊢ ( ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ℕ₀ )
322	320 321	syl	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ℕ₀ )
323	266	nnnn0d	⊢ ( 𝑚 ∈ ℕ₀ → ( 𝑚 + 1 ) ∈ ℕ₀ )
324	323	adantr	⊢ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) → ( 𝑚 + 1 ) ∈ ℕ₀ )
325	324	ad2antlr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ( 𝑚 + 1 ) ∈ ℕ₀ )
326		leloe	⊢ ( ( ( 𝑚 + 1 ) ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( 𝑚 + 1 ) ≤ 𝑁 ↔ ( ( 𝑚 + 1 ) < 𝑁 ∨ ( 𝑚 + 1 ) = 𝑁 ) ) )
327	272 8 326	syl2anr	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) → ( ( 𝑚 + 1 ) ≤ 𝑁 ↔ ( ( 𝑚 + 1 ) < 𝑁 ∨ ( 𝑚 + 1 ) = 𝑁 ) ) )
328	286 327	bitrd	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) → ( 𝑚 < 𝑁 ↔ ( ( 𝑚 + 1 ) < 𝑁 ∨ ( 𝑚 + 1 ) = 𝑁 ) ) )
329	328	biimpd	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) → ( 𝑚 < 𝑁 → ( ( 𝑚 + 1 ) < 𝑁 ∨ ( 𝑚 + 1 ) = 𝑁 ) ) )
330	329	imdistani	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑚 < 𝑁 ) → ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( ( 𝑚 + 1 ) < 𝑁 ∨ ( 𝑚 + 1 ) = 𝑁 ) ) )
331	330	anasss	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( ( 𝑚 + 1 ) < 𝑁 ∨ ( 𝑚 + 1 ) = 𝑁 ) ) )
332		simplll	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 𝜑 )
333	280	nnge1d	⊢ ( 𝑚 ∈ ℕ₀ → 1 ≤ ( ( 𝑚 + 1 ) + 1 ) )
334	333	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 1 ≤ ( ( 𝑚 + 1 ) + 1 ) )
335		zltp1le	⊢ ( ( ( 𝑚 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑚 + 1 ) < 𝑁 ↔ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) )
336	289 284 335	syl2anr	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) → ( ( 𝑚 + 1 ) < 𝑁 ↔ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) )
337	336	biimpa	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 )
338	289	peano2zd	⊢ ( 𝑚 ∈ ℕ₀ → ( ( 𝑚 + 1 ) + 1 ) ∈ ℤ )
339		1z	⊢ 1 ∈ ℤ
340		elfz	⊢ ( ( ( ( 𝑚 + 1 ) + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ↔ ( 1 ≤ ( ( 𝑚 + 1 ) + 1 ) ∧ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) ) )
341	339 340	mp3an2	⊢ ( ( ( ( 𝑚 + 1 ) + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ↔ ( 1 ≤ ( ( 𝑚 + 1 ) + 1 ) ∧ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) ) )
342	338 284 341	syl2anr	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) → ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ↔ ( 1 ≤ ( ( 𝑚 + 1 ) + 1 ) ∧ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) ) )
343	342	adantr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ↔ ( 1 ≤ ( ( 𝑚 + 1 ) + 1 ) ∧ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) ) )
344	334 337 343	mpbir2and	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) )
345	344	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) )
346		nn0re	⊢ ( 𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ )
347	346	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 𝑚 ∈ ℝ )
348	272	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( 𝑚 + 1 ) ∈ ℝ )
349	8	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 𝑁 ∈ ℝ )
350	346	ltp1d	⊢ ( 𝑚 ∈ ℕ₀ → 𝑚 < ( 𝑚 + 1 ) )
351	350	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 𝑚 < ( 𝑚 + 1 ) )
352		simpr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( 𝑚 + 1 ) < 𝑁 )
353	347 348 349 351 352	lttrd	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 𝑚 < 𝑁 )
354	353	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 𝑚 < 𝑁 )
355		anass	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑚 < 𝑁 ) ↔ ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) )
356	304	anassrs	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
357	355 356	sylanb	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑚 < 𝑁 ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
358	357	an32s	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ 𝑚 < 𝑁 ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
359	354 358	syldan	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
360		ffn	⊢ ( 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) → 𝑞 Fn ( 1 ... ( 𝑚 + 1 ) ) )
361	360	ad2antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 𝑞 Fn ( 1 ... ( 𝑚 + 1 ) ) )
362	275	ad3antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
363		eluz	⊢ ( ( ( ( 𝑚 + 1 ) + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ_≥ ‘ ( ( 𝑚 + 1 ) + 1 ) ) ↔ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) )
364	338 284 363	syl2anr	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) → ( 𝑁 ∈ ( ℤ_≥ ‘ ( ( 𝑚 + 1 ) + 1 ) ) ↔ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) )
365	364	adantr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( 𝑁 ∈ ( ℤ_≥ ‘ ( ( 𝑚 + 1 ) + 1 ) ) ↔ ( ( 𝑚 + 1 ) + 1 ) ≤ 𝑁 ) )
366	337 365	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → 𝑁 ∈ ( ℤ_≥ ‘ ( ( 𝑚 + 1 ) + 1 ) ) )
367		eluzfz1	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( ( 𝑚 + 1 ) + 1 ) ) → ( ( 𝑚 + 1 ) + 1 ) ∈ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) )
368	366 367	syl	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( 𝑚 + 1 ) + 1 ) ∈ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) )
369	368	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( 𝑚 + 1 ) + 1 ) ∈ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) )
370		fnconstg	⊢ ( 0 ∈ V → ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) Fn ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) )
371	75 370	ax-mp	⊢ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) Fn ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 )
372		fvun2	⊢ ( ( 𝑞 Fn ( 1 ... ( 𝑚 + 1 ) ) ∧ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) Fn ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ∧ ( ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ ∧ ( ( 𝑚 + 1 ) + 1 ) ∈ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) )
373	371 372	mp3an2	⊢ ( ( 𝑞 Fn ( 1 ... ( 𝑚 + 1 ) ) ∧ ( ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ ∧ ( ( 𝑚 + 1 ) + 1 ) ∈ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) )
374	361 362 369 373	syl12anc	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) )
375	75	fvconst2	⊢ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) → ( ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 )
376	369 375	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 )
377	374 376	eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 )
378		nfv	⊢ Ⅎ 𝑝 ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) )
379		nfcv	⊢ Ⅎ 𝑝 <
380		nfcv	⊢ Ⅎ 𝑝 ( ( 𝑚 + 1 ) + 1 )
381	306 379 380	nfbr	⊢ Ⅎ 𝑝 ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < ( ( 𝑚 + 1 ) + 1 )
382	378 381	nfim	⊢ Ⅎ 𝑝 ( ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < ( ( 𝑚 + 1 ) + 1 ) )
383		fveq1	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) )
384	383	eqeq1d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ↔ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) )
385	313 384	3anbi23d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ↔ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) )
386	385	anbi2d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) ↔ ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) ) )
387	315	breq1d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( 𝐵 < ( ( 𝑚 + 1 ) + 1 ) ↔ ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < ( ( 𝑚 + 1 ) + 1 ) ) )
388	386 387	imbi12d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) → 𝐵 < ( ( 𝑚 + 1 ) + 1 ) ) ↔ ( ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < ( ( 𝑚 + 1 ) + 1 ) ) ) )
389		ovex	⊢ ( ( 𝑚 + 1 ) + 1 ) ∈ V
390		eleq1	⊢ ( 𝑛 = ( ( 𝑚 + 1 ) + 1 ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↔ ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ) )
391		fveqeq2	⊢ ( 𝑛 = ( ( 𝑚 + 1 ) + 1 ) → ( ( 𝑝 ‘ 𝑛 ) = 0 ↔ ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) )
392	390 391	3anbi13d	⊢ ( 𝑛 = ( ( 𝑚 + 1 ) + 1 ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ↔ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) )
393	392	anbi2d	⊢ ( 𝑛 = ( ( 𝑚 + 1 ) + 1 ) → ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ) ↔ ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) ) )
394		breq2	⊢ ( 𝑛 = ( ( 𝑚 + 1 ) + 1 ) → ( 𝐵 < 𝑛 ↔ 𝐵 < ( ( 𝑚 + 1 ) + 1 ) ) )
395	393 394	imbi12d	⊢ ( 𝑛 = ( ( 𝑚 + 1 ) + 1 ) → ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ) → 𝐵 < 𝑛 ) ↔ ( ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) → 𝐵 < ( ( 𝑚 + 1 ) + 1 ) ) ) )
396	389 395 4	vtocl	⊢ ( ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) → 𝐵 < ( ( 𝑚 + 1 ) + 1 ) )
397	382 312 388 396	vtoclf	⊢ ( ( 𝜑 ∧ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ ( ( 𝑚 + 1 ) + 1 ) ) = 0 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < ( ( 𝑚 + 1 ) + 1 ) )
398	332 345 359 377 397	syl13anc	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < ( ( 𝑚 + 1 ) + 1 ) )
399	355 320	sylanb	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑚 < 𝑁 ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) )
400	399	an32s	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ 𝑚 < 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) )
401	400	elfzelzd	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ 𝑚 < 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ℤ )
402	354 401	syldan	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ℤ )
403	289	ad3antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( 𝑚 + 1 ) ∈ ℤ )
404		zleltp1	⊢ ( ( ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ℤ ∧ ( 𝑚 + 1 ) ∈ ℤ ) → ( ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ ( 𝑚 + 1 ) ↔ ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < ( ( 𝑚 + 1 ) + 1 ) ) )
405	402 403 404	syl2anc	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ( ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ ( 𝑚 + 1 ) ↔ ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < ( ( 𝑚 + 1 ) + 1 ) ) )
406	398 405	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) < 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ ( 𝑚 + 1 ) )
407	350	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → 𝑚 < ( 𝑚 + 1 ) )
408		breq2	⊢ ( ( 𝑚 + 1 ) = 𝑁 → ( 𝑚 < ( 𝑚 + 1 ) ↔ 𝑚 < 𝑁 ) )
409	408	biimpac	⊢ ( ( 𝑚 < ( 𝑚 + 1 ) ∧ ( 𝑚 + 1 ) = 𝑁 ) → 𝑚 < 𝑁 )
410	407 409	sylan	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) = 𝑁 ) → 𝑚 < 𝑁 )
411		elfzle2	⊢ ( ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ 𝑁 )
412	400 411	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ 𝑚 < 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ 𝑁 )
413	410 412	syldan	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) = 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ 𝑁 )
414		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) = 𝑁 ) → ( 𝑚 + 1 ) = 𝑁 )
415	413 414	breqtrrd	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( 𝑚 + 1 ) = 𝑁 ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ ( 𝑚 + 1 ) )
416	406 415	jaodan	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ∧ ( ( 𝑚 + 1 ) < 𝑁 ∨ ( 𝑚 + 1 ) = 𝑁 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ ( 𝑚 + 1 ) )
417	416	an32s	⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( ( 𝑚 + 1 ) < 𝑁 ∨ ( 𝑚 + 1 ) = 𝑁 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ ( 𝑚 + 1 ) )
418	331 417	sylan	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ ( 𝑚 + 1 ) )
419		elfz2nn0	⊢ ( ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... ( 𝑚 + 1 ) ) ↔ ( ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ℕ₀ ∧ ( 𝑚 + 1 ) ∈ ℕ₀ ∧ ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≤ ( 𝑚 + 1 ) ) )
420	322 325 418 419	syl3anbrc	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∈ ( 0 ... ( 𝑚 + 1 ) ) )
421		fzss2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑚 + 1 ) ) → ( 1 ... ( 𝑚 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
422	293 421	syl	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 1 ... ( 𝑚 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
423	422	sselda	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ) → 𝑛 ∈ ( 1 ... 𝑁 ) )
424	423	3ad2antr1	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 0 ) ) → 𝑛 ∈ ( 1 ... 𝑁 ) )
425	356	3ad2antr2	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 0 ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
426	360	ad2antll	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → 𝑞 Fn ( 1 ... ( 𝑚 + 1 ) ) )
427	275	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
428		simprl	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) )
429		fvun1	⊢ ( ( 𝑞 Fn ( 1 ... ( 𝑚 + 1 ) ) ∧ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) Fn ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ∧ ( ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ ∧ 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = ( 𝑞 ‘ 𝑛 ) )
430	371 429	mp3an2	⊢ ( ( 𝑞 Fn ( 1 ... ( 𝑚 + 1 ) ) ∧ ( ( ( 1 ... ( 𝑚 + 1 ) ) ∩ ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ ∧ 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = ( 𝑞 ‘ 𝑛 ) )
431	426 427 428 430	syl12anc	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = ( 𝑞 ‘ 𝑛 ) )
432	431	adantlrr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = ( 𝑞 ‘ 𝑛 ) )
433	432	3adantr3	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 0 ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = ( 𝑞 ‘ 𝑛 ) )
434		simpr3	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 0 ) ) → ( 𝑞 ‘ 𝑛 ) = 0 )
435	433 434	eqtrd	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 0 ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 )
436	424 425 435	3jca	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 0 ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) )
437		nfv	⊢ Ⅎ 𝑝 ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) )
438		nfcv	⊢ Ⅎ 𝑝 𝑛
439	306 379 438	nfbr	⊢ Ⅎ 𝑝 ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < 𝑛
440	437 439	nfim	⊢ Ⅎ 𝑝 ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < 𝑛 )
441		fveq1	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( 𝑝 ‘ 𝑛 ) = ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) )
442	441	eqeq1d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝑝 ‘ 𝑛 ) = 0 ↔ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) )
443	313 442	3anbi23d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) ) )
444	443	anbi2d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ) ↔ ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) ) ) )
445	315	breq1d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( 𝐵 < 𝑛 ↔ ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < 𝑛 ) )
446	444 445	imbi12d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 0 ) ) → 𝐵 < 𝑛 ) ↔ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < 𝑛 ) ) )
447	440 312 446 4	vtoclf	⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < 𝑛 )
448	447	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 0 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < 𝑛 )
449	436 448	syldan	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 0 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 < 𝑛 )
450		simp1	⊢ ( ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) → 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) )
451	423	anasss	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ) ) → 𝑛 ∈ ( 1 ... 𝑁 ) )
452	450 451	sylanr2	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) ) → 𝑛 ∈ ( 1 ... 𝑁 ) )
453		simp2	⊢ ( ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) → 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) )
454	453 304	sylanr2	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) ) → ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) )
455	431	3adantr3	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = ( 𝑞 ‘ 𝑛 ) )
456		simpr3	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) → ( 𝑞 ‘ 𝑛 ) = 𝐾 )
457	455 456	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ₀ ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 )
458	457	anasss	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 )
459	458	adantrlr	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) ) → ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 )
460	452 454 459	3jca	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 ) )
461		nfv	⊢ Ⅎ 𝑝 ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 ) )
462		nfcv	⊢ Ⅎ 𝑝 ( 𝑛 − 1 )
463	306 462	nfne	⊢ Ⅎ 𝑝 ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≠ ( 𝑛 − 1 )
464	461 463	nfim	⊢ Ⅎ 𝑝 ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≠ ( 𝑛 − 1 ) )
465	441	eqeq1d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝑝 ‘ 𝑛 ) = 𝐾 ↔ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 ) )
466	313 465	3anbi23d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝐾 ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 ) ) )
467	466	anbi2d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝐾 ) ) ↔ ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 ) ) ) )
468	315	neeq1d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( 𝐵 ≠ ( 𝑛 − 1 ) ↔ ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≠ ( 𝑛 − 1 ) ) )
469	467 468	imbi12d	⊢ ( 𝑝 = ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) → ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑝 : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑝 ‘ 𝑛 ) = 𝐾 ) ) → 𝐵 ≠ ( 𝑛 − 1 ) ) ↔ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≠ ( 𝑛 − 1 ) ) ) )
470	464 312 469 5	vtoclf	⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ... 𝐾 ) ∧ ( ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) ‘ 𝑛 ) = 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≠ ( 𝑛 − 1 ) )
471	460 470	syldan	⊢ ( ( 𝜑 ∧ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≠ ( 𝑛 − 1 ) )
472	471	anassrs	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝑛 ∈ ( 1 ... ( 𝑚 + 1 ) ) ∧ 𝑞 : ( 1 ... ( 𝑚 + 1 ) ) ⟶ ( 0 ... 𝐾 ) ∧ ( 𝑞 ‘ 𝑛 ) = 𝐾 ) ) → ⦋ ( 𝑞 ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ≠ ( 𝑛 − 1 ) )
473	267 269 420 449 472	poimirlem27	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 2 ∥ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) )
474	267 269 420	poimirlem26	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 2 ∥ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
475		fzfi	⊢ ( 0 ... ( 𝑚 + 1 ) ) ∈ Fin
476		xpfi	⊢ ( ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∈ Fin ∧ ( 0 ... ( 𝑚 + 1 ) ) ∈ Fin ) → ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∈ Fin )
477	256 475 476	mp2an	⊢ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∈ Fin
478		rabfi	⊢ ( ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∈ Fin → { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∈ Fin )
479		hashcl	⊢ ( { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∈ Fin → ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℕ₀ )
480	477 478 479	mp2b	⊢ ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℕ₀
481	480	nn0zi	⊢ ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℤ
482		zsubcl	⊢ ( ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℤ ∧ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ∈ ℤ ) → ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) ∈ ℤ )
483	481 264 482	mp2an	⊢ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) ∈ ℤ
484		zsubcl	⊢ ( ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℤ ∧ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℤ ) → ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ∈ ℤ )
485	481 260 484	mp2an	⊢ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ∈ ℤ
486		dvds2sub	⊢ ( ( 2 ∈ ℤ ∧ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) ∈ ℤ ∧ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ∈ ℤ ) → ( ( 2 ∥ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) ∧ 2 ∥ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) → 2 ∥ ( ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) − ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) ) )
487	242 483 485 486	mp3an	⊢ ( ( 2 ∥ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) ∧ 2 ∥ ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) → 2 ∥ ( ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) − ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) )
488	473 474 487	syl2anc	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 2 ∥ ( ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) − ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) )
489	480	nn0cni	⊢ ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℂ
490	263	nn0cni	⊢ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ∈ ℂ
491	259	nn0cni	⊢ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℂ
492		nnncan1	⊢ ( ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℂ ∧ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ∈ ℂ ∧ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℂ ) → ( ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) − ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) = ( ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) )
493	489 490 491 492	mp3an	⊢ ( ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) − ( ( ♯ ‘ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) × ( 0 ... ( 𝑚 + 1 ) ) ) ∣ ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( ( 0 ... ( 𝑚 + 1 ) ) ∖ { ( 2^nd ‘ 𝑡 ) } ) 𝑖 = ⦋ ( 1^st ‘ 𝑡 ) / 𝑠 ⦌ ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) = ( ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) )
494	488 493	breqtrdi	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 2 ∥ ( ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) )
495		dvdssub2	⊢ ( ( ( 2 ∈ ℤ ∧ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ∈ ℤ ∧ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ∈ ℤ ) ∧ 2 ∥ ( ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) − ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) ) → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) )
496	265 494 495	sylancr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ) )
497		nn0cn	⊢ ( 𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ )
498		pncan1	⊢ ( 𝑚 ∈ ℂ → ( ( 𝑚 + 1 ) − 1 ) = 𝑚 )
499	497 498	syl	⊢ ( 𝑚 ∈ ℕ₀ → ( ( 𝑚 + 1 ) − 1 ) = 𝑚 )
500	499	oveq2d	⊢ ( 𝑚 ∈ ℕ₀ → ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) = ( 0 ... 𝑚 ) )
501	500	rexeqdv	⊢ ( 𝑚 ∈ ℕ₀ → ( ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
502	500 501	raleqbidv	⊢ ( 𝑚 ∈ ℕ₀ → ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ) )
503	502	3anbi1d	⊢ ( 𝑚 ∈ ℕ₀ → ( ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) ↔ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) ) )
504	503	rabbidv	⊢ ( 𝑚 ∈ ℕ₀ → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } = { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } )
505	504	fveq2d	⊢ ( 𝑚 ∈ ℕ₀ → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) )
506	505	ad2antrl	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) )
507	1	adantr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 𝑁 ∈ ℕ )
508	6	adantr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 𝐾 ∈ ℕ )
509		simprl	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 𝑚 ∈ ℕ₀ )
510		simprr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → 𝑚 < 𝑁 )
511	507 508 509 510	poimirlem4	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ≈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } )
512		fzfi	⊢ ( 1 ... 𝑚 ) ∈ Fin
513		mapfi	⊢ ( ( ( 0 ..^ 𝐾 ) ∈ Fin ∧ ( 1 ... 𝑚 ) ∈ Fin ) → ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) ∈ Fin )
514	15 512 513	mp2an	⊢ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) ∈ Fin
515		ovex	⊢ ( 1 ... 𝑚 ) ∈ V
516	515 515	mapval	⊢ ( ( 1 ... 𝑚 ) ↑_m ( 1 ... 𝑚 ) ) = { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) ⟶ ( 1 ... 𝑚 ) }
517		mapfi	⊢ ( ( ( 1 ... 𝑚 ) ∈ Fin ∧ ( 1 ... 𝑚 ) ∈ Fin ) → ( ( 1 ... 𝑚 ) ↑_m ( 1 ... 𝑚 ) ) ∈ Fin )
518	512 512 517	mp2an	⊢ ( ( 1 ... 𝑚 ) ↑_m ( 1 ... 𝑚 ) ) ∈ Fin
519	516 518	eqeltrri	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) ⟶ ( 1 ... 𝑚 ) } ∈ Fin
520		f1of	⊢ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) → 𝑓 : ( 1 ... 𝑚 ) ⟶ ( 1 ... 𝑚 ) )
521	520	ss2abi	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ⊆ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) ⟶ ( 1 ... 𝑚 ) }
522		ssfi	⊢ ( ( { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) ⟶ ( 1 ... 𝑚 ) } ∈ Fin ∧ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ⊆ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) ⟶ ( 1 ... 𝑚 ) } ) → { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ∈ Fin )
523	519 521 522	mp2an	⊢ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ∈ Fin
524		xpfi	⊢ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) ∈ Fin ∧ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ∈ Fin ) → ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∈ Fin )
525	514 523 524	mp2an	⊢ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∈ Fin
526		rabfi	⊢ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∈ Fin → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∈ Fin )
527	525 526	ax-mp	⊢ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∈ Fin
528		rabfi	⊢ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∈ Fin → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ∈ Fin )
529	256 528	ax-mp	⊢ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ∈ Fin
530		hashen	⊢ ( ( { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ∈ Fin ∧ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ∈ Fin ) → ( ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ↔ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ≈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) )
531	527 529 530	mp2an	⊢ ( ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ↔ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ≈ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } )
532	511 531	sylibr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) )
533	506 532	eqtr4d	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) )
534	533	breq2d	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ( ∀ 𝑖 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( ( 𝑚 + 1 ) − 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ∧ ( ( 1^st ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = 0 ∧ ( ( 2^nd ‘ 𝑠 ) ‘ ( 𝑚 + 1 ) ) = ( 𝑚 + 1 ) ) } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
535	496 534	bitrd	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
536	535	biimpd	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) → 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
537	536	con3d	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) ) → ( ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
538	537	expcom	⊢ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) → ( 𝜑 → ( ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) )
539	538	a2d	⊢ ( ( 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) → ( ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) → ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) )
540	539	3adant1	⊢ ( ( 𝑁 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀ ∧ 𝑚 < 𝑁 ) → ( ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑚 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ ( 1 ... 𝑚 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑚 ) ∃ 𝑗 ∈ ( 0 ... 𝑚 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑚 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑚 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) → ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... ( 𝑚 + 1 ) ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... ( 𝑚 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑚 + 1 ) ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... ( 𝑚 + 1 ) ) ∃ 𝑗 ∈ ( 0 ... ( 𝑚 + 1 ) ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... ( 𝑚 + 1 ) ) ) × { 0 } ) ) ) ∪ ( ( ( ( 𝑚 + 1 ) + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) ) )
541	112 137 162 187 241 540	fnn0ind	⊢ ( ( 𝑁 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑁 ≤ 𝑁 ) → ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
542	10 541	mpcom	⊢ ( 𝜑 → ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) )
543		dvds0	⊢ ( 2 ∈ ℤ → 2 ∥ 0 )
544	242 543	ax-mp	⊢ 2 ∥ 0
545		hash0	⊢ ( ♯ ‘ ∅ ) = 0
546	544 545	breqtrri	⊢ 2 ∥ ( ♯ ‘ ∅ )
547		fveq2	⊢ ( { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } = ∅ → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } ) = ( ♯ ‘ ∅ ) )
548	546 547	breqtrrid	⊢ ( { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } = ∅ → 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } ) )
549	8	ltp1d	⊢ ( 𝜑 → 𝑁 < ( 𝑁 + 1 ) )
550	284	peano2zd	⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℤ )
551		fzn	⊢ ( ( ( 𝑁 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 < ( 𝑁 + 1 ) ↔ ( ( 𝑁 + 1 ) ... 𝑁 ) = ∅ ) )
552	550 284 551	syl2anc	⊢ ( 𝜑 → ( 𝑁 < ( 𝑁 + 1 ) ↔ ( ( 𝑁 + 1 ) ... 𝑁 ) = ∅ ) )
553	549 552	mpbid	⊢ ( 𝜑 → ( ( 𝑁 + 1 ) ... 𝑁 ) = ∅ )
554	553	xpeq1d	⊢ ( 𝜑 → ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) = ( ∅ × { 0 } ) )
555	554 91	eqtrdi	⊢ ( 𝜑 → ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) = ∅ )
556	555	uneq2d	⊢ ( 𝜑 → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ∅ ) )
557		un0	⊢ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ∅ ) = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
558	556 557	eqtrdi	⊢ ( 𝜑 → ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) = ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
559	558	csbeq1d	⊢ ( 𝜑 → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = ⦋ ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) / 𝑝 ⦌ 𝐵 )
560		ovex	⊢ ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∈ V
561	560 2	csbie	⊢ ⦋ ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) / 𝑝 ⦌ 𝐵 = 𝐶
562	559 561	eqtrdi	⊢ ( 𝜑 → ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 = 𝐶 )
563	562	eqeq2d	⊢ ( 𝜑 → ( 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ 𝑖 = 𝐶 ) )
564	563	rexbidv	⊢ ( 𝜑 → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 ) )
565	564	ralbidv	⊢ ( 𝜑 → ( ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 ↔ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 ) )
566	565	rabbidv	⊢ ( 𝜑 → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } = { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } )
567	566	fveq2d	⊢ ( 𝜑 → ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) = ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } ) )
568	567	breq2d	⊢ ( 𝜑 → ( 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ↔ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } ) ) )
569	548 568	syl5ibr	⊢ ( 𝜑 → ( { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } = ∅ → 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) ) )
570	569	necon3bd	⊢ ( 𝜑 → ( ¬ 2 ∥ ( ♯ ‘ { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = ⦋ ( ( ( 1^st ‘ 𝑠 ) ∘_f + ( ( ( ( 2^nd ‘ 𝑠 ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ 𝑠 ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∪ ( ( ( 𝑁 + 1 ) ... 𝑁 ) × { 0 } ) ) / 𝑝 ⦌ 𝐵 } ) → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } ≠ ∅ ) )
571	542 570	mpd	⊢ ( 𝜑 → { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } ≠ ∅ )
572		rabn0	⊢ ( { 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∣ ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 } ≠ ∅ ↔ ∃ 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 )
573	571 572	sylib	⊢ ( 𝜑 → ∃ 𝑠 ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑖 = 𝐶 )

Metamath Proof Explorer

Theorem poimirlem28

Proof