subfacp1lem5 - Mathbox for Mario Carneiro

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Theorem subfacp1lem5 33155

Description: Lemma for subfacp1 33157. In subfacp1lem6 33156 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements with (𝑓‘(𝑓‘1)) ≠ 1 for fixed 𝑀 = (𝑓‘1) is in bijection with derangements of 2...(𝑁 + 1), because pre-composing with the function 𝐹 swaps 1 and 𝑀 and turns the function into a bijection with (𝑓‘1) = 1 and (𝑓‘𝑥) ≠ 𝑥 for all other 𝑥, so dropping the point at 1 yields a derangement on the 𝑁 remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.)

**Hypotheses**
Ref	Expression
derang.d	⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
subfac.n	⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ (𝐷‘(1...𝑛)))
subfacp1lem.a	⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}
subfacp1lem1.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
subfacp1lem1.m	⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))
subfacp1lem1.x	⊢ 𝑀 ∈ V
subfacp1lem1.k	⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
subfacp1lem5.b	⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)}
subfacp1lem5.f	⊢ 𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})
subfacp1lem5.c	⊢ 𝐶 = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}

**Assertion**
Ref	Expression
subfacp1lem5	⊢ (𝜑 → (♯‘𝐵) = (𝑆‘𝑁))

Distinct variable groups: 𝑓,𝑔,𝑛,𝑥,𝑦,𝐴 𝑓,𝐹,𝑔,𝑥,𝑦 𝑓,𝑁,𝑔,𝑛,𝑥,𝑦 𝐵,𝑓,𝑔,𝑥,𝑦 𝑥,𝐶,𝑦 𝜑,𝑥,𝑦 𝐷,𝑛 𝑓,𝐾,𝑛,𝑥,𝑦 𝑓,𝑀,𝑔,𝑥,𝑦 𝑆,𝑛,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑓,𝑔,𝑛) 𝐵(𝑛) 𝐶(𝑓,𝑔,𝑛) 𝐷(𝑥,𝑦,𝑓,𝑔) 𝑆(𝑓,𝑔) 𝐹(𝑛) 𝐾(𝑔) 𝑀(𝑛)

Proof of Theorem subfacp1lem5 Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subfacp1lem.a	. . . . . . 7 ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}
2		fzfi 13701	. . . . . . . 8 ⊢ (1...(𝑁 + 1)) ∈ Fin
3		deranglem 33137	. . . . . . . 8 ⊢ ((1...(𝑁 + 1)) ∈ Fin → {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin
5	1, 4	eqeltri 2836	. . . . . 6 ⊢ 𝐴 ∈ Fin
6		subfacp1lem5.b	. . . . . . 7 ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)}
7	6	ssrab3 4016	. . . . . 6 ⊢ 𝐵 ⊆ 𝐴
8		ssfi 8965	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
9	5, 7, 8	mp2an 689	. . . . 5 ⊢ 𝐵 ∈ Fin
10	9	elexi 3452	. . . 4 ⊢ 𝐵 ∈ V
11	10	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
12		eqid 2739	. . . 4 ⊢ (𝑏 ∈ 𝐵 ↦ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))) = (𝑏 ∈ 𝐵 ↦ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))))
13		derang.d	. . . . . . . . . . 11 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
14		subfac.n	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ (𝐷‘(1...𝑛)))
15		subfacp1lem1.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
16		subfacp1lem1.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))
17		subfacp1lem1.x	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
18		subfacp1lem1.k	. . . . . . . . . . 11 ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
19		subfacp1lem5.f	. . . . . . . . . . 11 ⊢ 𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})
20		f1oi 6763	. . . . . . . . . . . 12 ⊢ ( I ↾ 𝐾):𝐾–1-1-onto→𝐾
21	20	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ( I ↾ 𝐾):𝐾–1-1-onto→𝐾)
22	13, 14, 1, 15, 16, 17, 18, 19, 21	subfacp1lem2a 33151	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1))
23	22	simp1d 1141	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
24		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
25		fveq1 6782	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑏 → (𝑔‘1) = (𝑏‘1))
26	25	eqeq1d 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑏 → ((𝑔‘1) = 𝑀 ↔ (𝑏‘1) = 𝑀))
27		fveq1 6782	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑏 → (𝑔‘𝑀) = (𝑏‘𝑀))
28	27	neeq1d 3004	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑏 → ((𝑔‘𝑀) ≠ 1 ↔ (𝑏‘𝑀) ≠ 1))
29	26, 28	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑏 → (((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1) ↔ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) ≠ 1)))
30	29, 6	elrab2 3628	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) ≠ 1)))
31	24, 30	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) ≠ 1)))
32	31	simpld 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐴)
33		vex 3437	. . . . . . . . . . . 12 ⊢ 𝑏 ∈ V
34		f1oeq1 6713	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑏 → (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
35		fveq1 6782	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑏 → (𝑓‘𝑦) = (𝑏‘𝑦))
36	35	neeq1d 3004	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑏 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦))
37	36	ralbidv 3113	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑏 → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))
38	34, 37	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) ↔ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)))
39	33, 38, 1	elab2 3614	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝐴 ↔ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))
40	32, 39	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))
41	40	simpld 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
42		f1oco 6748	. . . . . . . . 9 ⊢ ((𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) → (𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
43	23, 41, 42	syl2an2r 682	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
44		f1of1 6724	. . . . . . . 8 ⊢ ((𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)))
45		df-f1 6442	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) ↔ ((𝐹 ∘ 𝑏):(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ Fun ^◡(𝐹 ∘ 𝑏)))
46	45	simprbi 497	. . . . . . . 8 ⊢ ((𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) → Fun ^◡(𝐹 ∘ 𝑏))
47	43, 44, 46	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Fun ^◡(𝐹 ∘ 𝑏))
48		f1ofn 6726	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝐹 ∘ 𝑏) Fn (1...(𝑁 + 1)))
49		fnresdm 6560	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝑏) Fn (1...(𝑁 + 1)) → ((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))) = (𝐹 ∘ 𝑏))
50		f1oeq1 6713	. . . . . . . . . 10 ⊢ (((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))) = (𝐹 ∘ 𝑏) → (((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ (𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
51	43, 48, 49, 50	4syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ (𝐹 ∘ 𝑏):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
52	43, 51	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
53		f1ofo 6732	. . . . . . . 8 ⊢ (((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → ((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1)))
54	52, 53	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1)))
55		1ex 10980	. . . . . . . . . 10 ⊢ 1 ∈ V
56	55, 55	f1osn 6765	. . . . . . . . 9 ⊢ {⟨1, 1⟩}:{1}–1-1-onto→{1}
57	43, 48	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹 ∘ 𝑏) Fn (1...(𝑁 + 1)))
58	15	peano2nnd 11999	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
59		nnuz 12630	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
60	58, 59	eleqtrdi 2850	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘1))
61		eluzfz1 13272	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑁 + 1)))
62	60, 61	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ (1...(𝑁 + 1)))
63	62	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 1 ∈ (1...(𝑁 + 1)))
64		fnressn 7039	. . . . . . . . . . . 12 ⊢ (((𝐹 ∘ 𝑏) Fn (1...(𝑁 + 1)) ∧ 1 ∈ (1...(𝑁 + 1))) → ((𝐹 ∘ 𝑏) ↾ {1}) = {⟨1, ((𝐹 ∘ 𝑏)‘1)⟩})
65	57, 63, 64	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ {1}) = {⟨1, ((𝐹 ∘ 𝑏)‘1)⟩})
66		f1of 6725	. . . . . . . . . . . . . . . 16 ⊢ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
67	41, 66	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
68	67, 63	fvco3d 6877	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏)‘1) = (𝐹‘(𝑏‘1)))
69	31	simprd 496	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) ≠ 1))
70	69	simpld 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘1) = 𝑀)
71	70	fveq2d 6787	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝑏‘1)) = (𝐹‘𝑀))
72	22	simp3d 1143	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑀) = 1)
73	72	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑀) = 1)
74	68, 71, 73	3eqtrd 2783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏)‘1) = 1)
75	74	opeq2d 4812	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ⟨1, ((𝐹 ∘ 𝑏)‘1)⟩ = ⟨1, 1⟩)
76	75	sneqd 4574	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → {⟨1, ((𝐹 ∘ 𝑏)‘1)⟩} = {⟨1, 1⟩})
77	65, 76	eqtrd 2779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ {1}) = {⟨1, 1⟩})
78	77	f1oeq1d 6720	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (((𝐹 ∘ 𝑏) ↾ {1}):{1}–1-1-onto→{1} ↔ {⟨1, 1⟩}:{1}–1-1-onto→{1}))
79	56, 78	mpbiri 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ {1}):{1}–1-1-onto→{1})
80		f1ofo 6732	. . . . . . . 8 ⊢ (((𝐹 ∘ 𝑏) ↾ {1}):{1}–1-1-onto→{1} → ((𝐹 ∘ 𝑏) ↾ {1}):{1}–onto→{1})
81	79, 80	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ {1}):{1}–onto→{1})
82		resdif 6746	. . . . . . 7 ⊢ ((Fun ^◡(𝐹 ∘ 𝑏) ∧ ((𝐹 ∘ 𝑏) ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1)) ∧ ((𝐹 ∘ 𝑏) ↾ {1}):{1}–onto→{1}) → ((𝐹 ∘ 𝑏) ↾ ((1...(𝑁 + 1)) ∖ {1})):((1...(𝑁 + 1)) ∖ {1})–1-1-onto→((1...(𝑁 + 1)) ∖ {1}))
83	47, 54, 81, 82	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ ((1...(𝑁 + 1)) ∖ {1})):((1...(𝑁 + 1)) ∖ {1})–1-1-onto→((1...(𝑁 + 1)) ∖ {1}))
84		fzsplit 13291	. . . . . . . . . . 11 ⊢ (1 ∈ (1...(𝑁 + 1)) → (1...(𝑁 + 1)) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
85	62, 84	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑁 + 1)) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
86		1z 12359	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
87		fzsn 13307	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → (1...1) = {1})
88	86, 87	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...1) = {1}
89		1p1e2 12107	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
90	89	oveq1i 7294	. . . . . . . . . . 11 ⊢ ((1 + 1)...(𝑁 + 1)) = (2...(𝑁 + 1))
91	88, 90	uneq12i 4096	. . . . . . . . . 10 ⊢ ((1...1) ∪ ((1 + 1)...(𝑁 + 1))) = ({1} ∪ (2...(𝑁 + 1)))
92	85, 91	eqtr2di 2796	. . . . . . . . 9 ⊢ (𝜑 → ({1} ∪ (2...(𝑁 + 1))) = (1...(𝑁 + 1)))
93	62	snssd 4743	. . . . . . . . . 10 ⊢ (𝜑 → {1} ⊆ (1...(𝑁 + 1)))
94		incom 4136	. . . . . . . . . . 11 ⊢ ({1} ∩ (2...(𝑁 + 1))) = ((2...(𝑁 + 1)) ∩ {1})
95		1lt2 12153	. . . . . . . . . . . . . 14 ⊢ 1 < 2
96		1re 10984	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
97		2re 12056	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
98	96, 97	ltnlei 11105	. . . . . . . . . . . . . 14 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
99	95, 98	mpbi 229	. . . . . . . . . . . . 13 ⊢ ¬ 2 ≤ 1
100		elfzle1 13268	. . . . . . . . . . . . 13 ⊢ (1 ∈ (2...(𝑁 + 1)) → 2 ≤ 1)
101	99, 100	mto 196	. . . . . . . . . . . 12 ⊢ ¬ 1 ∈ (2...(𝑁 + 1))
102		disjsn 4648	. . . . . . . . . . . 12 ⊢ (((2...(𝑁 + 1)) ∩ {1}) = ∅ ↔ ¬ 1 ∈ (2...(𝑁 + 1)))
103	101, 102	mpbir 230	. . . . . . . . . . 11 ⊢ ((2...(𝑁 + 1)) ∩ {1}) = ∅
104	94, 103	eqtri 2767	. . . . . . . . . 10 ⊢ ({1} ∩ (2...(𝑁 + 1))) = ∅
105		uneqdifeq 4424	. . . . . . . . . 10 ⊢ (({1} ⊆ (1...(𝑁 + 1)) ∧ ({1} ∩ (2...(𝑁 + 1))) = ∅) → (({1} ∪ (2...(𝑁 + 1))) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1))))
106	93, 104, 105	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (({1} ∪ (2...(𝑁 + 1))) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1))))
107	92, 106	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1)))
108	107	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1)))
109		reseq2 5889	. . . . . . . . 9 ⊢ (((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1)) → ((𝐹 ∘ 𝑏) ↾ ((1...(𝑁 + 1)) ∖ {1})) = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))))
110	109	f1oeq1d 6720	. . . . . . . 8 ⊢ (((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1)) → (((𝐹 ∘ 𝑏) ↾ ((1...(𝑁 + 1)) ∖ {1})):((1...(𝑁 + 1)) ∖ {1})–1-1-onto→((1...(𝑁 + 1)) ∖ {1}) ↔ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):((1...(𝑁 + 1)) ∖ {1})–1-1-onto→((1...(𝑁 + 1)) ∖ {1})))
111		f1oeq2 6714	. . . . . . . 8 ⊢ (((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1)) → (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):((1...(𝑁 + 1)) ∖ {1})–1-1-onto→((1...(𝑁 + 1)) ∖ {1}) ↔ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→((1...(𝑁 + 1)) ∖ {1})))
112		f1oeq3 6715	. . . . . . . 8 ⊢ (((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1)) → (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→((1...(𝑁 + 1)) ∖ {1}) ↔ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1))))
113	110, 111, 112	3bitrd 305	. . . . . . 7 ⊢ (((1...(𝑁 + 1)) ∖ {1}) = (2...(𝑁 + 1)) → (((𝐹 ∘ 𝑏) ↾ ((1...(𝑁 + 1)) ∖ {1})):((1...(𝑁 + 1)) ∖ {1})–1-1-onto→((1...(𝑁 + 1)) ∖ {1}) ↔ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1))))
114	108, 113	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (((𝐹 ∘ 𝑏) ↾ ((1...(𝑁 + 1)) ∖ {1})):((1...(𝑁 + 1)) ∖ {1})–1-1-onto→((1...(𝑁 + 1)) ∖ {1}) ↔ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1))))
115	83, 114	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)))
116	67	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → 𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
117		fzp1ss 13316	. . . . . . . . . . 11 ⊢ (1 ∈ ℤ → ((1 + 1)...(𝑁 + 1)) ⊆ (1...(𝑁 + 1)))
118	86, 117	ax-mp 5	. . . . . . . . . 10 ⊢ ((1 + 1)...(𝑁 + 1)) ⊆ (1...(𝑁 + 1))
119	90, 118	eqsstrri 3957	. . . . . . . . 9 ⊢ (2...(𝑁 + 1)) ⊆ (1...(𝑁 + 1))
120		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → 𝑦 ∈ (2...(𝑁 + 1)))
121	119, 120	sselid 3920	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → 𝑦 ∈ (1...(𝑁 + 1)))
122	116, 121	fvco3d 6877	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → ((𝐹 ∘ 𝑏)‘𝑦) = (𝐹‘(𝑏‘𝑦)))
123	13, 14, 1, 15, 16, 17, 18, 6, 19	subfacp1lem4 33154	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝐹 = 𝐹)
124	123	fveq1d 6785	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹‘𝑦) = (𝐹‘𝑦))
125	124	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (^◡𝐹‘𝑦) = (𝐹‘𝑦))
126	69	simprd 496	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) ≠ 1)
127	126, 73	neeqtrrd 3019	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) ≠ (𝐹‘𝑀))
128	127	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑏‘𝑀) ≠ (𝐹‘𝑀))
129		fveq2 6783	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑀 → (𝑏‘𝑦) = (𝑏‘𝑀))
130		fveq2 6783	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑀 → (𝐹‘𝑦) = (𝐹‘𝑀))
131	129, 130	neeq12d 3006	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) ≠ (𝐹‘𝑦) ↔ (𝑏‘𝑀) ≠ (𝐹‘𝑀)))
132	128, 131	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑦 = 𝑀 → (𝑏‘𝑦) ≠ (𝐹‘𝑦)))
133	119	sseli 3918	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (2...(𝑁 + 1)) → 𝑦 ∈ (1...(𝑁 + 1)))
134	40	simprd 496	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)
135	134	r19.21bi 3135	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → (𝑏‘𝑦) ≠ 𝑦)
136	133, 135	sylan2 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑏‘𝑦) ≠ 𝑦)
137	136	adantrr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀)) → (𝑏‘𝑦) ≠ 𝑦)
138	18	eleq2i 2831	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐾 ↔ 𝑦 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}))
139		eldifsn 4721	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) ↔ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀))
140	138, 139	bitri 274	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐾 ↔ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀))
141	13, 14, 1, 15, 16, 17, 18, 19, 21	subfacp1lem2b 33152	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐹‘𝑦) = (( I ↾ 𝐾)‘𝑦))
142		fvresi 7054	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐾 → (( I ↾ 𝐾)‘𝑦) = 𝑦)
143	142	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (( I ↾ 𝐾)‘𝑦) = 𝑦)
144	141, 143	eqtrd 2779	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐹‘𝑦) = 𝑦)
145	140, 144	sylan2br 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀)) → (𝐹‘𝑦) = 𝑦)
146	145	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀)) → (𝐹‘𝑦) = 𝑦)
147	137, 146	neeqtrrd 3019	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀)) → (𝑏‘𝑦) ≠ (𝐹‘𝑦))
148	147	expr 457	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑦 ≠ 𝑀 → (𝑏‘𝑦) ≠ (𝐹‘𝑦)))
149	132, 148	pm2.61dne 3032	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑏‘𝑦) ≠ (𝐹‘𝑦))
150	149	necomd 3000	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝐹‘𝑦) ≠ (𝑏‘𝑦))
151	125, 150	eqnetrd 3012	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (^◡𝐹‘𝑦) ≠ (𝑏‘𝑦))
152	23	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
153		ffvelrn 6968	. . . . . . . . . . 11 ⊢ ((𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → (𝑏‘𝑦) ∈ (1...(𝑁 + 1)))
154	67, 133, 153	syl2an 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑏‘𝑦) ∈ (1...(𝑁 + 1)))
155		f1ocnvfv 7159	. . . . . . . . . 10 ⊢ ((𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝑏‘𝑦) ∈ (1...(𝑁 + 1))) → ((𝐹‘(𝑏‘𝑦)) = 𝑦 → (^◡𝐹‘𝑦) = (𝑏‘𝑦)))
156	152, 154, 155	syl2an2r 682	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → ((𝐹‘(𝑏‘𝑦)) = 𝑦 → (^◡𝐹‘𝑦) = (𝑏‘𝑦)))
157	156	necon3d 2965	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → ((^◡𝐹‘𝑦) ≠ (𝑏‘𝑦) → (𝐹‘(𝑏‘𝑦)) ≠ 𝑦))
158	151, 157	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝐹‘(𝑏‘𝑦)) ≠ 𝑦)
159	122, 158	eqnetrd 3012	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → ((𝐹 ∘ 𝑏)‘𝑦) ≠ 𝑦)
160	159	ralrimiva 3104	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) ≠ 𝑦)
161		f1of 6725	. . . . . . . . . . . 12 ⊢ (( I ↾ 𝐾):𝐾–1-1-onto→𝐾 → ( I ↾ 𝐾):𝐾⟶𝐾)
162	20, 161	ax-mp 5	. . . . . . . . . . 11 ⊢ ( I ↾ 𝐾):𝐾⟶𝐾
163		fzfi 13701	. . . . . . . . . . . . 13 ⊢ (2...(𝑁 + 1)) ∈ Fin
164		difexg 5252	. . . . . . . . . . . . 13 ⊢ ((2...(𝑁 + 1)) ∈ Fin → ((2...(𝑁 + 1)) ∖ {𝑀}) ∈ V)
165	163, 164	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((2...(𝑁 + 1)) ∖ {𝑀}) ∈ V
166	18, 165	eqeltri 2836	. . . . . . . . . . 11 ⊢ 𝐾 ∈ V
167		fex 7111	. . . . . . . . . . 11 ⊢ ((( I ↾ 𝐾):𝐾⟶𝐾 ∧ 𝐾 ∈ V) → ( I ↾ 𝐾) ∈ V)
168	162, 166, 167	mp2an 689	. . . . . . . . . 10 ⊢ ( I ↾ 𝐾) ∈ V
169		prex 5356	. . . . . . . . . 10 ⊢ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩} ∈ V
170	168, 169	unex 7605	. . . . . . . . 9 ⊢ (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ V
171	19, 170	eqeltri 2836	. . . . . . . 8 ⊢ 𝐹 ∈ V
172	171, 33	coex 7786	. . . . . . 7 ⊢ (𝐹 ∘ 𝑏) ∈ V
173	172	resex 5942	. . . . . 6 ⊢ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) ∈ V
174		f1oeq1 6713	. . . . . . 7 ⊢ (𝑓 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) → (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ↔ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1))))
175		fveq1 6782	. . . . . . . . . 10 ⊢ (𝑓 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) → (𝑓‘𝑦) = (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦))
176		fvres 6802	. . . . . . . . . 10 ⊢ (𝑦 ∈ (2...(𝑁 + 1)) → (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦) = ((𝐹 ∘ 𝑏)‘𝑦))
177	175, 176	sylan9eq 2799	. . . . . . . . 9 ⊢ ((𝑓 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑓‘𝑦) = ((𝐹 ∘ 𝑏)‘𝑦))
178	177	neeq1d 3004	. . . . . . . 8 ⊢ ((𝑓 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → ((𝑓‘𝑦) ≠ 𝑦 ↔ ((𝐹 ∘ 𝑏)‘𝑦) ≠ 𝑦))
179	178	ralbidva 3112	. . . . . . 7 ⊢ (𝑓 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) → (∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) ≠ 𝑦))
180	174, 179	anbi12d 631	. . . . . 6 ⊢ (𝑓 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) → ((𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) ↔ (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) ≠ 𝑦)))
181		subfacp1lem5.c	. . . . . 6 ⊢ 𝐶 = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}
182	173, 180, 181	elab2 3614	. . . . 5 ⊢ (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) ∈ 𝐶 ↔ (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))):(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) ≠ 𝑦))
183	115, 160, 182	sylanbrc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) ∈ 𝐶)
184		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ 𝐶)
185		vex 3437	. . . . . . . . . . . 12 ⊢ 𝑐 ∈ V
186		f1oeq1 6713	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑐 → (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ↔ 𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1))))
187		fveq1 6782	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑐 → (𝑓‘𝑦) = (𝑐‘𝑦))
188	187	neeq1d 3004	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑐 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑐‘𝑦) ≠ 𝑦))
189	188	ralbidv 3113	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑐 → (∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑐‘𝑦) ≠ 𝑦))
190	186, 189	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑐 → ((𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) ↔ (𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑐‘𝑦) ≠ 𝑦)))
191	185, 190, 181	elab2 3614	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐶 ↔ (𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑐‘𝑦) ≠ 𝑦))
192	184, 191	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑐‘𝑦) ≠ 𝑦))
193	192	simpld 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)))
194		f1oun 6744	. . . . . . . . . 10 ⊢ ((({⟨1, 1⟩}:{1}–1-1-onto→{1} ∧ 𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1))) ∧ (({1} ∩ (2...(𝑁 + 1))) = ∅ ∧ ({1} ∩ (2...(𝑁 + 1))) = ∅)) → ({⟨1, 1⟩} ∪ 𝑐):({1} ∪ (2...(𝑁 + 1)))–1-1-onto→({1} ∪ (2...(𝑁 + 1))))
195	104, 104, 194	mpanr12 702	. . . . . . . . 9 ⊢ (({⟨1, 1⟩}:{1}–1-1-onto→{1} ∧ 𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1))) → ({⟨1, 1⟩} ∪ 𝑐):({1} ∪ (2...(𝑁 + 1)))–1-1-onto→({1} ∪ (2...(𝑁 + 1))))
196	56, 193, 195	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ({⟨1, 1⟩} ∪ 𝑐):({1} ∪ (2...(𝑁 + 1)))–1-1-onto→({1} ∪ (2...(𝑁 + 1))))
197		f1oeq2 6714	. . . . . . . . . . 11 ⊢ (({1} ∪ (2...(𝑁 + 1))) = (1...(𝑁 + 1)) → (({⟨1, 1⟩} ∪ 𝑐):({1} ∪ (2...(𝑁 + 1)))–1-1-onto→({1} ∪ (2...(𝑁 + 1))) ↔ ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→({1} ∪ (2...(𝑁 + 1)))))
198		f1oeq3 6715	. . . . . . . . . . 11 ⊢ (({1} ∪ (2...(𝑁 + 1))) = (1...(𝑁 + 1)) → (({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→({1} ∪ (2...(𝑁 + 1))) ↔ ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
199	197, 198	bitrd 278	. . . . . . . . . 10 ⊢ (({1} ∪ (2...(𝑁 + 1))) = (1...(𝑁 + 1)) → (({⟨1, 1⟩} ∪ 𝑐):({1} ∪ (2...(𝑁 + 1)))–1-1-onto→({1} ∪ (2...(𝑁 + 1))) ↔ ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
200	92, 199	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (({⟨1, 1⟩} ∪ 𝑐):({1} ∪ (2...(𝑁 + 1)))–1-1-onto→({1} ∪ (2...(𝑁 + 1))) ↔ ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
201	200	biimpa 477	. . . . . . . 8 ⊢ ((𝜑 ∧ ({⟨1, 1⟩} ∪ 𝑐):({1} ∪ (2...(𝑁 + 1)))–1-1-onto→({1} ∪ (2...(𝑁 + 1)))) → ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
202	196, 201	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
203		f1oco 6748	. . . . . . 7 ⊢ ((𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) → (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
204	23, 202, 203	syl2an2r 682	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
205		f1of 6725	. . . . . . . . . 10 ⊢ (({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
206	202, 205	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
207		fvco3 6876	. . . . . . . . 9 ⊢ ((({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦) = (𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
208	206, 207	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦) = (𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
209	124	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → (^◡𝐹‘𝑦) = (𝐹‘𝑦))
210		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → 𝑦 ∈ (1...(𝑁 + 1)))
211	92	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → ({1} ∪ (2...(𝑁 + 1))) = (1...(𝑁 + 1)))
212	210, 211	eleqtrrd 2843	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → 𝑦 ∈ ({1} ∪ (2...(𝑁 + 1))))
213		elun 4084	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ({1} ∪ (2...(𝑁 + 1))) ↔ (𝑦 ∈ {1} ∨ 𝑦 ∈ (2...(𝑁 + 1))))
214	212, 213	sylib 217	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → (𝑦 ∈ {1} ∨ 𝑦 ∈ (2...(𝑁 + 1))))
215		nelne2 3043	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ (2...(𝑁 + 1)) ∧ ¬ 1 ∈ (2...(𝑁 + 1))) → 𝑀 ≠ 1)
216	16, 101, 215	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ≠ 1)
217	216	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ≠ 1)
218	22	simp2d 1142	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹‘1) = 𝑀)
219	218	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐹‘1) = 𝑀)
220		f1ofun 6727	. . . . . . . . . . . . . . . . . 18 ⊢ (({⟨1, 1⟩} ∪ 𝑐):({1} ∪ (2...(𝑁 + 1)))–1-1-onto→({1} ∪ (2...(𝑁 + 1))) → Fun ({⟨1, 1⟩} ∪ 𝑐))
221	196, 220	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → Fun ({⟨1, 1⟩} ∪ 𝑐))
222		ssun1 4107	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨1, 1⟩} ⊆ ({⟨1, 1⟩} ∪ 𝑐)
223	55	snid 4598	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ {1}
224	55	dmsnop 6124	. . . . . . . . . . . . . . . . . . 19 ⊢ dom {⟨1, 1⟩} = {1}
225	223, 224	eleqtrri 2839	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ dom {⟨1, 1⟩}
226		funssfv 6804	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun ({⟨1, 1⟩} ∪ 𝑐) ∧ {⟨1, 1⟩} ⊆ ({⟨1, 1⟩} ∪ 𝑐) ∧ 1 ∈ dom {⟨1, 1⟩}) → (({⟨1, 1⟩} ∪ 𝑐)‘1) = ({⟨1, 1⟩}‘1))
227	222, 225, 226	mp3an23 1452	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ({⟨1, 1⟩} ∪ 𝑐) → (({⟨1, 1⟩} ∪ 𝑐)‘1) = ({⟨1, 1⟩}‘1))
228	221, 227	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (({⟨1, 1⟩} ∪ 𝑐)‘1) = ({⟨1, 1⟩}‘1))
229	55, 55	fvsn 7062	. . . . . . . . . . . . . . . 16 ⊢ ({⟨1, 1⟩}‘1) = 1
230	228, 229	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (({⟨1, 1⟩} ∪ 𝑐)‘1) = 1)
231	217, 219, 230	3netr4d 3022	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐹‘1) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘1))
232		elsni 4579	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {1} → 𝑦 = 1)
233	232	fveq2d 6787	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {1} → (𝐹‘𝑦) = (𝐹‘1))
234	232	fveq2d 6787	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {1} → (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘1))
235	233, 234	neeq12d 3006	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {1} → ((𝐹‘𝑦) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ (𝐹‘1) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘1)))
236	231, 235	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑦 ∈ {1} → (𝐹‘𝑦) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
237	236	imp 407	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ {1}) → (𝐹‘𝑦) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦))
238	221	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → Fun ({⟨1, 1⟩} ∪ 𝑐))
239		ssun2 4108	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ⊆ ({⟨1, 1⟩} ∪ 𝑐)
240	239	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → 𝑐 ⊆ ({⟨1, 1⟩} ∪ 𝑐))
241		f1odm 6729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) → dom 𝑐 = (2...(𝑁 + 1)))
242	193, 241	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → dom 𝑐 = (2...(𝑁 + 1)))
243	242	eleq2d 2825	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑦 ∈ dom 𝑐 ↔ 𝑦 ∈ (2...(𝑁 + 1))))
244	243	biimpar 478	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → 𝑦 ∈ dom 𝑐)
245		funssfv 6804	. . . . . . . . . . . . . . 15 ⊢ ((Fun ({⟨1, 1⟩} ∪ 𝑐) ∧ 𝑐 ⊆ ({⟨1, 1⟩} ∪ 𝑐) ∧ 𝑦 ∈ dom 𝑐) → (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) = (𝑐‘𝑦))
246	238, 240, 244, 245	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) = (𝑐‘𝑦))
247		f1of 6725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) → 𝑐:(2...(𝑁 + 1))⟶(2...(𝑁 + 1)))
248	193, 247	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐:(2...(𝑁 + 1))⟶(2...(𝑁 + 1)))
249	16	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ∈ (2...(𝑁 + 1)))
250	248, 249	ffvelrnd 6971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐‘𝑀) ∈ (2...(𝑁 + 1)))
251		nelne2 3043	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐‘𝑀) ∈ (2...(𝑁 + 1)) ∧ ¬ 1 ∈ (2...(𝑁 + 1))) → (𝑐‘𝑀) ≠ 1)
252	250, 101, 251	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐‘𝑀) ≠ 1)
253	252	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑐‘𝑀) ≠ 1)
254	72	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝐹‘𝑀) = 1)
255	253, 254	neeqtrrd 3019	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑐‘𝑀) ≠ (𝐹‘𝑀))
256		fveq2 6783	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑀 → (𝑐‘𝑦) = (𝑐‘𝑀))
257	256, 130	neeq12d 3006	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑀 → ((𝑐‘𝑦) ≠ (𝐹‘𝑦) ↔ (𝑐‘𝑀) ≠ (𝐹‘𝑀)))
258	255, 257	syl5ibrcom 246	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑦 = 𝑀 → (𝑐‘𝑦) ≠ (𝐹‘𝑦)))
259	192	simprd 496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (2...(𝑁 + 1))(𝑐‘𝑦) ≠ 𝑦)
260	259	r19.21bi 3135	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑐‘𝑦) ≠ 𝑦)
261	260	adantrr 714	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀)) → (𝑐‘𝑦) ≠ 𝑦)
262	145	adantlr 712	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀)) → (𝐹‘𝑦) = 𝑦)
263	261, 262	neeqtrrd 3019	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ (𝑦 ∈ (2...(𝑁 + 1)) ∧ 𝑦 ≠ 𝑀)) → (𝑐‘𝑦) ≠ (𝐹‘𝑦))
264	263	expr 457	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑦 ≠ 𝑀 → (𝑐‘𝑦) ≠ (𝐹‘𝑦)))
265	258, 264	pm2.61dne 3032	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝑐‘𝑦) ≠ (𝐹‘𝑦))
266	246, 265	eqnetrd 3012	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ≠ (𝐹‘𝑦))
267	266	necomd 3000	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (𝐹‘𝑦) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦))
268	237, 267	jaodan 955	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ (𝑦 ∈ {1} ∨ 𝑦 ∈ (2...(𝑁 + 1)))) → (𝐹‘𝑦) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦))
269	214, 268	syldan 591	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → (𝐹‘𝑦) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦))
270	209, 269	eqnetrd 3012	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → (^◡𝐹‘𝑦) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦))
271	23	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
272	206	ffvelrnda 6970	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ∈ (1...(𝑁 + 1)))
273		f1ocnvfv 7159	. . . . . . . . . . 11 ⊢ ((𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ∈ (1...(𝑁 + 1))) → ((𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘𝑦)) = 𝑦 → (^◡𝐹‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
274	271, 272, 273	syl2an2r 682	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → ((𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘𝑦)) = 𝑦 → (^◡𝐹‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
275	274	necon3d 2965	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → ((^◡𝐹‘𝑦) ≠ (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) → (𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘𝑦)) ≠ 𝑦))
276	270, 275	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → (𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘𝑦)) ≠ 𝑦)
277	208, 276	eqnetrd 3012	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ (1...(𝑁 + 1))) → ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦) ≠ 𝑦)
278	277	ralrimiva 3104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (1...(𝑁 + 1))((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦) ≠ 𝑦)
279		snex 5355	. . . . . . . . 9 ⊢ {⟨1, 1⟩} ∈ V
280	279, 185	unex 7605	. . . . . . . 8 ⊢ ({⟨1, 1⟩} ∪ 𝑐) ∈ V
281	171, 280	coex 7786	. . . . . . 7 ⊢ (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ∈ V
282		f1oeq1 6713	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
283		fveq1 6782	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → (𝑓‘𝑦) = ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦))
284	283	neeq1d 3004	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → ((𝑓‘𝑦) ≠ 𝑦 ↔ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦) ≠ 𝑦))
285	284	ralbidv 3113	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦) ≠ 𝑦))
286	282, 285	anbi12d 631	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) ↔ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦) ≠ 𝑦)))
287	281, 286, 1	elab2 3614	. . . . . 6 ⊢ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ∈ 𝐴 ↔ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑦) ≠ 𝑦))
288	204, 278, 287	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ∈ 𝐴)
289	62	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 1 ∈ (1...(𝑁 + 1)))
290	206, 289	fvco3d 6877	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘1) = (𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘1)))
291	230	fveq2d 6787	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘1)) = (𝐹‘1))
292	290, 291, 219	3eqtrd 2783	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘1) = 𝑀)
293	119, 16	sselid 3920	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 + 1)))
294	293	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ∈ (1...(𝑁 + 1)))
295	206, 294	fvco3d 6877	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑀) = (𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘𝑀)))
296	239	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ⊆ ({⟨1, 1⟩} ∪ 𝑐))
297	249, 242	eleqtrrd 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ∈ dom 𝑐)
298		funssfv 6804	. . . . . . . . . 10 ⊢ ((Fun ({⟨1, 1⟩} ∪ 𝑐) ∧ 𝑐 ⊆ ({⟨1, 1⟩} ∪ 𝑐) ∧ 𝑀 ∈ dom 𝑐) → (({⟨1, 1⟩} ∪ 𝑐)‘𝑀) = (𝑐‘𝑀))
299	221, 296, 297, 298	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (({⟨1, 1⟩} ∪ 𝑐)‘𝑀) = (𝑐‘𝑀))
300	299	fveq2d 6787	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐹‘(({⟨1, 1⟩} ∪ 𝑐)‘𝑀)) = (𝐹‘(𝑐‘𝑀)))
301	295, 300	eqtrd 2779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑀) = (𝐹‘(𝑐‘𝑀)))
302	123	fveq1d 6785	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹‘1) = (𝐹‘1))
303	302, 218	eqtrd 2779	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹‘1) = 𝑀)
304	303	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (^◡𝐹‘1) = 𝑀)
305		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀)
306	256, 305	neeq12d 3006	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑀 → ((𝑐‘𝑦) ≠ 𝑦 ↔ (𝑐‘𝑀) ≠ 𝑀))
307	306, 259, 249	rspcdva 3563	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐‘𝑀) ≠ 𝑀)
308	307	necomd 3000	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ≠ (𝑐‘𝑀))
309	304, 308	eqnetrd 3012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (^◡𝐹‘1) ≠ (𝑐‘𝑀))
310	119, 250	sselid 3920	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐‘𝑀) ∈ (1...(𝑁 + 1)))
311		f1ocnvfv 7159	. . . . . . . . . 10 ⊢ ((𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝑐‘𝑀) ∈ (1...(𝑁 + 1))) → ((𝐹‘(𝑐‘𝑀)) = 1 → (^◡𝐹‘1) = (𝑐‘𝑀)))
312	23, 310, 311	syl2an2r 682	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝐹‘(𝑐‘𝑀)) = 1 → (^◡𝐹‘1) = (𝑐‘𝑀)))
313	312	necon3d 2965	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((^◡𝐹‘1) ≠ (𝑐‘𝑀) → (𝐹‘(𝑐‘𝑀)) ≠ 1))
314	309, 313	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐹‘(𝑐‘𝑀)) ≠ 1)
315	301, 314	eqnetrd 3012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑀) ≠ 1)
316	292, 315	jca 512	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘1) = 𝑀 ∧ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑀) ≠ 1))
317		fveq1 6782	. . . . . . . 8 ⊢ (𝑔 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → (𝑔‘1) = ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘1))
318	317	eqeq1d 2741	. . . . . . 7 ⊢ (𝑔 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → ((𝑔‘1) = 𝑀 ↔ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘1) = 𝑀))
319		fveq1 6782	. . . . . . . 8 ⊢ (𝑔 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → (𝑔‘𝑀) = ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑀))
320	319	neeq1d 3004	. . . . . . 7 ⊢ (𝑔 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → ((𝑔‘𝑀) ≠ 1 ↔ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑀) ≠ 1))
321	318, 320	anbi12d 631	. . . . . 6 ⊢ (𝑔 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) → (((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1) ↔ (((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘1) = 𝑀 ∧ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑀) ≠ 1)))
322	321, 6	elrab2 3628	. . . . 5 ⊢ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ∈ 𝐵 ↔ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ∈ 𝐴 ∧ (((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘1) = 𝑀 ∧ ((𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))‘𝑀) ≠ 1)))
323	288, 316, 322	sylanbrc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ∈ 𝐵)
324	23	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
325		f1of1 6724	. . . . . . 7 ⊢ (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝐹:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)))
326	324, 325	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝐹:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)))
327		f1of 6725	. . . . . . . 8 ⊢ (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝐹:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
328	324, 327	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝐹:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
329	67	adantrr 714	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
330	328, 329	fcod 6635	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝐹 ∘ 𝑏):(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
331	206	adantrl 713	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
332		cocan1 7172	. . . . . 6 ⊢ ((𝐹:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) ∧ (𝐹 ∘ 𝑏):(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))⟶(1...(𝑁 + 1))) → ((𝐹 ∘ (𝐹 ∘ 𝑏)) = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ↔ (𝐹 ∘ 𝑏) = ({⟨1, 1⟩} ∪ 𝑐)))
333	326, 330, 331, 332	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ (𝐹 ∘ 𝑏)) = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ↔ (𝐹 ∘ 𝑏) = ({⟨1, 1⟩} ∪ 𝑐)))
334		coass 6173	. . . . . . 7 ⊢ ((𝐹 ∘ 𝐹) ∘ 𝑏) = (𝐹 ∘ (𝐹 ∘ 𝑏))
335	123	coeq1d 5773	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹 ∘ 𝐹) = (𝐹 ∘ 𝐹))
336		f1ococnv1 6754	. . . . . . . . . . . 12 ⊢ (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (1...(𝑁 + 1))))
337	23, 336	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹 ∘ 𝐹) = ( I ↾ (1...(𝑁 + 1))))
338	335, 337	eqtr3d 2781	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ 𝐹) = ( I ↾ (1...(𝑁 + 1))))
339	338	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝐹 ∘ 𝐹) = ( I ↾ (1...(𝑁 + 1))))
340	339	coeq1d 5773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ 𝐹) ∘ 𝑏) = (( I ↾ (1...(𝑁 + 1))) ∘ 𝑏))
341		fcoi2 6658	. . . . . . . . 9 ⊢ (𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) → (( I ↾ (1...(𝑁 + 1))) ∘ 𝑏) = 𝑏)
342	329, 341	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (( I ↾ (1...(𝑁 + 1))) ∘ 𝑏) = 𝑏)
343	340, 342	eqtrd 2779	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ 𝐹) ∘ 𝑏) = 𝑏)
344	334, 343	eqtr3id 2793	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝐹 ∘ (𝐹 ∘ 𝑏)) = 𝑏)
345	344	eqeq1d 2741	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ (𝐹 ∘ 𝑏)) = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ↔ 𝑏 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐))))
346	74	adantrr 714	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ 𝑏)‘1) = 1)
347	230	adantrl 713	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (({⟨1, 1⟩} ∪ 𝑐)‘1) = 1)
348	346, 347	eqtr4d 2782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ 𝑏)‘1) = (({⟨1, 1⟩} ∪ 𝑐)‘1))
349		fveq2 6783	. . . . . . . . . . . . 13 ⊢ (𝑦 = 1 → ((𝐹 ∘ 𝑏)‘𝑦) = ((𝐹 ∘ 𝑏)‘1))
350		fveq2 6783	. . . . . . . . . . . . 13 ⊢ (𝑦 = 1 → (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘1))
351	349, 350	eqeq12d 2755	. . . . . . . . . . . 12 ⊢ (𝑦 = 1 → (((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ ((𝐹 ∘ 𝑏)‘1) = (({⟨1, 1⟩} ∪ 𝑐)‘1)))
352	55, 351	ralsn 4618	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {1} ((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ ((𝐹 ∘ 𝑏)‘1) = (({⟨1, 1⟩} ∪ 𝑐)‘1))
353	348, 352	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ∀𝑦 ∈ {1} ((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦))
354	353	biantrurd 533	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ (∀𝑦 ∈ {1} ((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦))))
355		ralunb 4126	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ({1} ∪ (2...(𝑁 + 1)))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ (∀𝑦 ∈ {1} ((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
356	354, 355	bitr4di 289	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ ∀𝑦 ∈ ({1} ∪ (2...(𝑁 + 1)))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
357	176	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦) = ((𝐹 ∘ 𝑏)‘𝑦))
358	357	eqcomd 2745	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → ((𝐹 ∘ 𝑏)‘𝑦) = (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦))
359	246	adantlrl 717	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) = (𝑐‘𝑦))
360	358, 359	eqeq12d 2755	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑦 ∈ (2...(𝑁 + 1))) → (((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦) = (𝑐‘𝑦)))
361	360	ralbidva 3112	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (2...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦) = (𝑐‘𝑦)))
362	92	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ({1} ∪ (2...(𝑁 + 1))) = (1...(𝑁 + 1)))
363	362	raleqdv 3349	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ ({1} ∪ (2...(𝑁 + 1)))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
364	356, 361, 363	3bitr3rd 310	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (1...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦) ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦) = (𝑐‘𝑦)))
365	57	adantrr 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝐹 ∘ 𝑏) Fn (1...(𝑁 + 1)))
366	202	adantrl 713	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
367		f1ofn 6726	. . . . . . . . 9 ⊢ (({⟨1, 1⟩} ∪ 𝑐):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → ({⟨1, 1⟩} ∪ 𝑐) Fn (1...(𝑁 + 1)))
368	366, 367	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ({⟨1, 1⟩} ∪ 𝑐) Fn (1...(𝑁 + 1)))
369		eqfnfv 6918	. . . . . . . 8 ⊢ (((𝐹 ∘ 𝑏) Fn (1...(𝑁 + 1)) ∧ ({⟨1, 1⟩} ∪ 𝑐) Fn (1...(𝑁 + 1))) → ((𝐹 ∘ 𝑏) = ({⟨1, 1⟩} ∪ 𝑐) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
370	365, 368, 369	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ 𝑏) = ({⟨1, 1⟩} ∪ 𝑐) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝐹 ∘ 𝑏)‘𝑦) = (({⟨1, 1⟩} ∪ 𝑐)‘𝑦)))
371		fnssres 6564	. . . . . . . . 9 ⊢ (((𝐹 ∘ 𝑏) Fn (1...(𝑁 + 1)) ∧ (2...(𝑁 + 1)) ⊆ (1...(𝑁 + 1))) → ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) Fn (2...(𝑁 + 1)))
372	365, 119, 371	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) Fn (2...(𝑁 + 1)))
373	193	adantrl 713	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)))
374		f1ofn 6726	. . . . . . . . 9 ⊢ (𝑐:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) → 𝑐 Fn (2...(𝑁 + 1)))
375	373, 374	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 Fn (2...(𝑁 + 1)))
376		eqfnfv 6918	. . . . . . . 8 ⊢ ((((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) Fn (2...(𝑁 + 1)) ∧ 𝑐 Fn (2...(𝑁 + 1))) → (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) = 𝑐 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦) = (𝑐‘𝑦)))
377	372, 375, 376	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) = 𝑐 ↔ ∀𝑦 ∈ (2...(𝑁 + 1))(((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))‘𝑦) = (𝑐‘𝑦)))
378	364, 370, 377	3bitr4d 311	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ 𝑏) = ({⟨1, 1⟩} ∪ 𝑐) ↔ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) = 𝑐))
379		eqcom 2746	. . . . . 6 ⊢ (((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))) = 𝑐 ↔ 𝑐 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1))))
380	378, 379	bitrdi 287	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝐹 ∘ 𝑏) = ({⟨1, 1⟩} ∪ 𝑐) ↔ 𝑐 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))))
381	333, 345, 380	3bitr3d 309	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝐹 ∘ ({⟨1, 1⟩} ∪ 𝑐)) ↔ 𝑐 = ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))))
382	12, 183, 323, 381	f1o2d 7532	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ ((𝐹 ∘ 𝑏) ↾ (2...(𝑁 + 1)))):𝐵–1-1-onto→𝐶)
383	11, 382	hasheqf1od 14077	. 2 ⊢ (𝜑 → (♯‘𝐵) = (♯‘𝐶))
384	13, 14	derangen2 33145	. . . . 5 ⊢ ((2...(𝑁 + 1)) ∈ Fin → (𝐷‘(2...(𝑁 + 1))) = (𝑆‘(♯‘(2...(𝑁 + 1)))))
385	13	derangval 33138	. . . . . 6 ⊢ ((2...(𝑁 + 1)) ∈ Fin → (𝐷‘(2...(𝑁 + 1))) = (♯‘{𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}))
386	181	fveq2i 6786	. . . . . 6 ⊢ (♯‘𝐶) = (♯‘{𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)})
387	385, 386	eqtr4di 2797	. . . . 5 ⊢ ((2...(𝑁 + 1)) ∈ Fin → (𝐷‘(2...(𝑁 + 1))) = (♯‘𝐶))
388	384, 387	eqtr3d 2781	. . . 4 ⊢ ((2...(𝑁 + 1)) ∈ Fin → (𝑆‘(♯‘(2...(𝑁 + 1)))) = (♯‘𝐶))
389	163, 388	ax-mp 5	. . 3 ⊢ (𝑆‘(♯‘(2...(𝑁 + 1)))) = (♯‘𝐶)
390	15, 59	eleqtrdi 2850	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
391		eluzp1p1 12619	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → (𝑁 + 1) ∈ (ℤ_≥‘(1 + 1)))
392	390, 391	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘(1 + 1)))
393		df-2 12045	. . . . . . . 8 ⊢ 2 = (1 + 1)
394	393	fveq2i 6786	. . . . . . 7 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
395	392, 394	eleqtrrdi 2851	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘2))
396		hashfz 14151	. . . . . 6 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘2) → (♯‘(2...(𝑁 + 1))) = (((𝑁 + 1) − 2) + 1))
397	395, 396	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(2...(𝑁 + 1))) = (((𝑁 + 1) − 2) + 1))
398	58	nncnd 11998	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
399		2cnd 12060	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
400		1cnd 10979	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
401	398, 399, 400	subsubd 11369	. . . . 5 ⊢ (𝜑 → ((𝑁 + 1) − (2 − 1)) = (((𝑁 + 1) − 2) + 1))
402		2m1e1 12108	. . . . . . 7 ⊢ (2 − 1) = 1
403	402	oveq2i 7295	. . . . . 6 ⊢ ((𝑁 + 1) − (2 − 1)) = ((𝑁 + 1) − 1)
404	15	nncnd 11998	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
405		ax-1cn 10938	. . . . . . 7 ⊢ 1 ∈ ℂ
406		pncan 11236	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
407	404, 405, 406	sylancl 586	. . . . . 6 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
408	403, 407	eqtrid 2791	. . . . 5 ⊢ (𝜑 → ((𝑁 + 1) − (2 − 1)) = 𝑁)
409	397, 401, 408	3eqtr2d 2785	. . . 4 ⊢ (𝜑 → (♯‘(2...(𝑁 + 1))) = 𝑁)
410	409	fveq2d 6787	. . 3 ⊢ (𝜑 → (𝑆‘(♯‘(2...(𝑁 + 1)))) = (𝑆‘𝑁))
411	389, 410	eqtr3id 2793	. 2 ⊢ (𝜑 → (♯‘𝐶) = (𝑆‘𝑁))
412	383, 411	eqtrd 2779	1 ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2107 {cab 2716 ≠ wne 2944 ∀wral 3065 {crab 3069 Vcvv 3433 ∖ cdif 3885 ∪ cun 3886 ∩ cin 3887 ⊆ wss 3888 ∅c0 4257 {csn 4562 {cpr 4564 ⟨cop 4568 class class class wbr 5075 ↦ cmpt 5158 I cid 5489 ^◡ccnv 5589 dom cdm 5590 ↾ cres 5592 ∘ ccom 5594 Fun wfun 6431 Fn wfn 6432 ⟶wf 6433 –1-1→wf1 6434 –onto→wfo 6435 –1-1-onto→wf1o 6436 ‘cfv 6437 (class class class)co 7284 Fincfn 8742 ℂcc 10878 1c1 10881 + caddc 10883 < clt 11018 ≤ cle 11019 − cmin 11214 ℕcn 11982 2c2 12037 ℕ₀cn0 12242 ℤcz 12328 ℤ_≥cuz 12591 ...cfz 13248 ♯chash 14053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-1st 7840 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-oadd 8310 df-er 8507 df-map 8626 df-pm 8627 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-dju 9668 df-card 9706 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-nn 11983 df-2 12045 df-n0 12243 df-xnn0 12315 df-z 12329 df-uz 12592 df-fz 13249 df-hash 14054

This theorem is referenced by: subfacp1lem6 33156

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