subfacp1lem3 - Mathbox for Mario Carneiro

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Theorem subfacp1lem3 33044

Description: Lemma for subfacp1 33048. In subfacp1lem6 33047 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 that satisfy this for fixed 𝑀 = (𝑓‘1) is in bijection with 𝑁 − 1 derangements, by simply dropping the 𝑥 = 1 and 𝑥 = 𝑀 points from the function to get a derangement on 𝐾 = (1...(𝑁 − 1)) ∖ {1, 𝑀}. (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})
subfacp1lem3.b	⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)}
subfacp1lem3.c	⊢ 𝐶 = {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}

**Assertion**
Ref	Expression
subfacp1lem3	⊢ (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1)))

Distinct variable groups: 𝑓,𝑔,𝑛,𝑥,𝑦,𝐴 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) 𝐾(𝑔) 𝑀(𝑛)

Proof of Theorem subfacp1lem3 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 13620	. . . . . . . 8 ⊢ (1...(𝑁 + 1)) ∈ Fin
3		deranglem 33028	. . . . . . . 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 2835	. . . . . 6 ⊢ 𝐴 ∈ Fin
6		subfacp1lem3.b	. . . . . . 7 ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)}
7	6	ssrab3 4011	. . . . . 6 ⊢ 𝐵 ⊆ 𝐴
8		ssfi 8918	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
9	5, 7, 8	mp2an 688	. . . . 5 ⊢ 𝐵 ∈ Fin
10	9	elexi 3441	. . . 4 ⊢ 𝐵 ∈ V
11	10	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
12		eqid 2738	. . . 4 ⊢ (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾)) = (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾))
13		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
14		fveq1 6755	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑏 → (𝑔‘1) = (𝑏‘1))
15	14	eqeq1d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑏 → ((𝑔‘1) = 𝑀 ↔ (𝑏‘1) = 𝑀))
16		fveq1 6755	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑏 → (𝑔‘𝑀) = (𝑏‘𝑀))
17	16	eqeq1d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑏 → ((𝑔‘𝑀) = 1 ↔ (𝑏‘𝑀) = 1))
18	15, 17	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑏 → (((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1) ↔ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1)))
19	18, 6	elrab2 3620	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1)))
20	13, 19	sylib 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1)))
21	20	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐴)
22		vex 3426	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
23		f1oeq1 6688	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
24		fveq1 6755	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑏 → (𝑓‘𝑦) = (𝑏‘𝑦))
25	24	neeq1d 3002	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑏 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦))
26	25	ralbidv 3120	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))
27	23, 26	anbi12d 630	. . . . . . . . . . 11 ⊢ (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)))
28	22, 27, 1	elab2 3606	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐴 ↔ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))
29	21, 28	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))
30	29	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
31		f1of1 6699	. . . . . . . 8 ⊢ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)))
32		df-f1 6423	. . . . . . . . 9 ⊢ (𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) ↔ (𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ Fun ^◡𝑏))
33	32	simprbi 496	. . . . . . . 8 ⊢ (𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) → Fun ^◡𝑏)
34	30, 31, 33	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Fun ^◡𝑏)
35		f1ofn 6701	. . . . . . . . . . 11 ⊢ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑏 Fn (1...(𝑁 + 1)))
36	30, 35	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 Fn (1...(𝑁 + 1)))
37		fnresdm 6535	. . . . . . . . . 10 ⊢ (𝑏 Fn (1...(𝑁 + 1)) → (𝑏 ↾ (1...(𝑁 + 1))) = 𝑏)
38		f1oeq1 6688	. . . . . . . . . 10 ⊢ ((𝑏 ↾ (1...(𝑁 + 1))) = 𝑏 → ((𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
39	36, 37, 38	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
40	30, 39	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
41		f1ofo 6707	. . . . . . . 8 ⊢ ((𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1)))
42	40, 41	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1)))
43		ssun2 4103	. . . . . . . . . . . 12 ⊢ {1, 𝑀} ⊆ (𝐾 ∪ {1, 𝑀})
44		derang.d	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
45		subfac.n	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ (𝐷‘(1...𝑛)))
46		subfacp1lem1.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47		subfacp1lem1.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))
48		subfacp1lem1.x	. . . . . . . . . . . . . 14 ⊢ 𝑀 ∈ V
49		subfacp1lem1.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
50	44, 45, 1, 46, 47, 48, 49	subfacp1lem1 33041	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))
51	50	simp2d 1141	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
52	43, 51	sseqtrid 3969	. . . . . . . . . . 11 ⊢ (𝜑 → {1, 𝑀} ⊆ (1...(𝑁 + 1)))
53	52	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → {1, 𝑀} ⊆ (1...(𝑁 + 1)))
54	36, 53	fnssresd 6540	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}) Fn {1, 𝑀})
55	20	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1))
56	55	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘1) = 𝑀)
57	48	prid2 4696	. . . . . . . . . . . 12 ⊢ 𝑀 ∈ {1, 𝑀}
58	56, 57	eqeltrdi 2847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘1) ∈ {1, 𝑀})
59	55	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) = 1)
60		1ex 10902	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
61	60	prid1 4695	. . . . . . . . . . . 12 ⊢ 1 ∈ {1, 𝑀}
62	59, 61	eqeltrdi 2847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) ∈ {1, 𝑀})
63		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝑏‘𝑥) = (𝑏‘1))
64	63	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((𝑏‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘1) ∈ {1, 𝑀}))
65		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑀 → (𝑏‘𝑥) = (𝑏‘𝑀))
66	65	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑀 → ((𝑏‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘𝑀) ∈ {1, 𝑀}))
67	60, 48, 64, 66	ralpr 4633	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀} ↔ ((𝑏‘1) ∈ {1, 𝑀} ∧ (𝑏‘𝑀) ∈ {1, 𝑀}))
68	58, 62, 67	sylanbrc 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀})
69		fvres 6775	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {1, 𝑀} → ((𝑏 ↾ {1, 𝑀})‘𝑥) = (𝑏‘𝑥))
70	69	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {1, 𝑀} → (((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘𝑥) ∈ {1, 𝑀}))
71	70	ralbiia 3089	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀} ↔ ∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀})
72	68, 71	sylibr 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀})
73		ffnfv 6974	. . . . . . . . 9 ⊢ ((𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀} ↔ ((𝑏 ↾ {1, 𝑀}) Fn {1, 𝑀} ∧ ∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀}))
74	54, 72, 73	sylanbrc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀})
75		fveqeq2 6765	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) = 1 ↔ (𝑏‘𝑀) = 1))
76	75	rspcev 3552	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ {1, 𝑀} ∧ (𝑏‘𝑀) = 1) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1)
77	57, 59, 76	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1)
78		fveqeq2 6765	. . . . . . . . . . . 12 ⊢ (𝑦 = 1 → ((𝑏‘𝑦) = 𝑀 ↔ (𝑏‘1) = 𝑀))
79	78	rspcev 3552	. . . . . . . . . . 11 ⊢ ((1 ∈ {1, 𝑀} ∧ (𝑏‘1) = 𝑀) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀)
80	61, 56, 79	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀)
81		eqeq2 2750	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((𝑏‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 1))
82	81	rexbidv 3225	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1))
83		eqeq2 2750	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑀 → ((𝑏‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 𝑀))
84	83	rexbidv 3225	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀))
85	60, 48, 82, 84	ralpr 4633	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1 ∧ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀))
86	77, 80, 85	sylanbrc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥)
87		eqcom 2745	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ((𝑏 ↾ {1, 𝑀})‘𝑦) = 𝑥)
88		fvres 6775	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {1, 𝑀} → ((𝑏 ↾ {1, 𝑀})‘𝑦) = (𝑏‘𝑦))
89	88	eqeq1d 2740	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {1, 𝑀} → (((𝑏 ↾ {1, 𝑀})‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 𝑥))
90	87, 89	syl5bb 282	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {1, 𝑀} → (𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ (𝑏‘𝑦) = 𝑥))
91	90	rexbiia 3176	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥)
92	91	ralbii 3090	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥)
93	86, 92	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦))
94		dffo3 6960	. . . . . . . 8 ⊢ ((𝑏 ↾ {1, 𝑀}):{1, 𝑀}–onto→{1, 𝑀} ↔ ((𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀} ∧ ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦)))
95	74, 93, 94	sylanbrc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}):{1, 𝑀}–onto→{1, 𝑀})
96		resdif 6720	. . . . . . 7 ⊢ ((Fun ^◡𝑏 ∧ (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1)) ∧ (𝑏 ↾ {1, 𝑀}):{1, 𝑀}–onto→{1, 𝑀}) → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}))
97	34, 42, 95, 96	syl3anc 1369	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}))
98		uncom 4083	. . . . . . . . . 10 ⊢ ({1, 𝑀} ∪ 𝐾) = (𝐾 ∪ {1, 𝑀})
99	98, 51	syl5eq 2791	. . . . . . . . 9 ⊢ (𝜑 → ({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)))
100		incom 4131	. . . . . . . . . . 11 ⊢ ({1, 𝑀} ∩ 𝐾) = (𝐾 ∩ {1, 𝑀})
101	50	simp1d 1140	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∩ {1, 𝑀}) = ∅)
102	100, 101	syl5eq 2791	. . . . . . . . . 10 ⊢ (𝜑 → ({1, 𝑀} ∩ 𝐾) = ∅)
103		uneqdifeq 4420	. . . . . . . . . 10 ⊢ (({1, 𝑀} ⊆ (1...(𝑁 + 1)) ∧ ({1, 𝑀} ∩ 𝐾) = ∅) → (({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾))
104	52, 102, 103	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾))
105	99, 104	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾)
106	105	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾)
107		reseq2 5875	. . . . . . . . 9 ⊢ (((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾 → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})) = (𝑏 ↾ 𝐾))
108	107	f1oeq1d 6695	. . . . . . . 8 ⊢ (((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀})))
109		f1oeq2 6689	. . . . . . . 8 ⊢ (((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ 𝐾):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀})))
110		f1oeq3 6690	. . . . . . . 8 ⊢ (((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ 𝐾):𝐾–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾))
111	108, 109, 110	3bitrd 304	. . . . . . 7 ⊢ (((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾))
112	106, 111	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾))
113	97, 112	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾)
114		ssun1 4102	. . . . . . . 8 ⊢ 𝐾 ⊆ (𝐾 ∪ {1, 𝑀})
115	114, 51	sseqtrid 3969	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ (1...(𝑁 + 1)))
116	115	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐾 ⊆ (1...(𝑁 + 1)))
117	29	simprd 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)
118		ssralv 3983	. . . . . 6 ⊢ (𝐾 ⊆ (1...(𝑁 + 1)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦 → ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦))
119	116, 117, 118	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦)
120	22	resex 5928	. . . . . 6 ⊢ (𝑏 ↾ 𝐾) ∈ V
121		f1oeq1 6688	. . . . . . 7 ⊢ (𝑓 = (𝑏 ↾ 𝐾) → (𝑓:𝐾–1-1-onto→𝐾 ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾))
122		fveq1 6755	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏 ↾ 𝐾) → (𝑓‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦))
123		fvres 6775	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐾 → ((𝑏 ↾ 𝐾)‘𝑦) = (𝑏‘𝑦))
124	122, 123	sylan9eq 2799	. . . . . . . . 9 ⊢ ((𝑓 = (𝑏 ↾ 𝐾) ∧ 𝑦 ∈ 𝐾) → (𝑓‘𝑦) = (𝑏‘𝑦))
125	124	neeq1d 3002	. . . . . . . 8 ⊢ ((𝑓 = (𝑏 ↾ 𝐾) ∧ 𝑦 ∈ 𝐾) → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦))
126	125	ralbidva 3119	. . . . . . 7 ⊢ (𝑓 = (𝑏 ↾ 𝐾) → (∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦))
127	121, 126	anbi12d 630	. . . . . 6 ⊢ (𝑓 = (𝑏 ↾ 𝐾) → ((𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦) ↔ ((𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦)))
128		subfacp1lem3.c	. . . . . 6 ⊢ 𝐶 = {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}
129	120, 127, 128	elab2 3606	. . . . 5 ⊢ ((𝑏 ↾ 𝐾) ∈ 𝐶 ↔ ((𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦))
130	113, 119, 129	sylanbrc 582	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ 𝐾) ∈ 𝐶)
131	46	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑁 ∈ ℕ)
132	47	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ∈ (2...(𝑁 + 1)))
133		eqid 2738	. . . . . . . 8 ⊢ (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})
134		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ 𝐶)
135		vex 3426	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
136		f1oeq1 6688	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑐 → (𝑓:𝐾–1-1-onto→𝐾 ↔ 𝑐:𝐾–1-1-onto→𝐾))
137		fveq1 6755	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑐 → (𝑓‘𝑦) = (𝑐‘𝑦))
138	137	neeq1d 3002	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑐 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑐‘𝑦) ≠ 𝑦))
139	138	ralbidv 3120	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑐 → (∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦))
140	136, 139	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑐 → ((𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦) ↔ (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦)))
141	135, 140, 128	elab2 3606	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐶 ↔ (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦))
142	134, 141	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦))
143	142	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐:𝐾–1-1-onto→𝐾)
144	44, 45, 1, 131, 132, 48, 49, 133, 143	subfacp1lem2a 33042	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀 ∧ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1))
145	144	simp1d 1140	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
146	44, 45, 1, 131, 132, 48, 49, 133, 143	subfacp1lem2b 33043	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) = (𝑐‘𝑦))
147	142	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦)
148	147	r19.21bi 3132	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → (𝑐‘𝑦) ≠ 𝑦)
149	146, 148	eqnetrd 3010	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
150	149	ralrimiva 3107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ 𝐾 ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
151	144	simp2d 1141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀)
152		elfzuz 13181	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (2...(𝑁 + 1)) → 𝑀 ∈ (ℤ_≥‘2))
153		eluz2b3 12591	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
154	153	simprbi 496	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ≠ 1)
155	47, 152, 154	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≠ 1)
156	155	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ≠ 1)
157	151, 156	eqnetrd 3010	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) ≠ 1)
158	144	simp3d 1142	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1)
159	156	necomd 2998	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 1 ≠ 𝑀)
160	158, 159	eqnetrd 3010	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) ≠ 𝑀)
161		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1))
162		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → 𝑦 = 1)
163	161, 162	neeq12d 3004	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) ≠ 1))
164		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑀 → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀))
165		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀)
166	164, 165	neeq12d 3004	. . . . . . . . . 10 ⊢ (𝑦 = 𝑀 → (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) ≠ 𝑀))
167	60, 48, 163, 166	ralpr 4633	. . . . . . . . 9 ⊢ (∀𝑦 ∈ {1, 𝑀} ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦 ↔ (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) ≠ 1 ∧ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) ≠ 𝑀))
168	157, 160, 167	sylanbrc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ {1, 𝑀} ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
169		ralunb 4121	. . . . . . . 8 ⊢ (∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦 ↔ (∀𝑦 ∈ 𝐾 ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦 ∧ ∀𝑦 ∈ {1, 𝑀} ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦))
170	150, 168, 169	sylanbrc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
171	51	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
172	171	raleqdv 3339	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦))
173	170, 172	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
174		prex 5350	. . . . . . . 8 ⊢ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩} ∈ V
175	135, 174	unex 7574	. . . . . . 7 ⊢ (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ V
176		f1oeq1 6688	. . . . . . . 8 ⊢ (𝑓 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))))
177		fveq1 6755	. . . . . . . . . 10 ⊢ (𝑓 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (𝑓‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦))
178	177	neeq1d 3002	. . . . . . . . 9 ⊢ (𝑓 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → ((𝑓‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦))
179	178	ralbidv 3120	. . . . . . . 8 ⊢ (𝑓 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦))
180	176, 179	anbi12d 630	. . . . . . 7 ⊢ (𝑓 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)))
181	175, 180, 1	elab2 3606	. . . . . 6 ⊢ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ 𝐴 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦))
182	145, 173, 181	sylanbrc 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ 𝐴)
183	151, 158	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀 ∧ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1))
184		fveq1 6755	. . . . . . . 8 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (𝑔‘1) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1))
185	184	eqeq1d 2740	. . . . . . 7 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → ((𝑔‘1) = 𝑀 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀))
186		fveq1 6755	. . . . . . . 8 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (𝑔‘𝑀) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀))
187	186	eqeq1d 2740	. . . . . . 7 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → ((𝑔‘𝑀) = 1 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1))
188	185, 187	anbi12d 630	. . . . . 6 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1) ↔ (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀 ∧ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1)))
189	188, 6	elrab2 3620	. . . . 5 ⊢ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ 𝐵 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ 𝐴 ∧ (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀 ∧ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1)))
190	182, 183, 189	sylanbrc 582	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ 𝐵)
191	56	adantrr 713	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘1) = 𝑀)
192	151	adantrl 712	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀)
193	191, 192	eqtr4d 2781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘1) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1))
194	59	adantrr 713	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘𝑀) = 1)
195	158	adantrl 712	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1)
196	194, 195	eqtr4d 2781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘𝑀) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀))
197		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑦 = 1 → (𝑏‘𝑦) = (𝑏‘1))
198	197, 161	eqeq12d 2754	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → ((𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ (𝑏‘1) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1)))
199		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → (𝑏‘𝑦) = (𝑏‘𝑀))
200	199, 164	eqeq12d 2754	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ (𝑏‘𝑀) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀)))
201	60, 48, 198, 200	ralpr 4633	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ((𝑏‘1) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) ∧ (𝑏‘𝑀) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀)))
202	193, 196, 201	sylanbrc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦))
203	202	biantrud 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ∧ ∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦))))
204		ralunb 4121	. . . . . . . 8 ⊢ (∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ∧ ∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦)))
205	203, 204	bitr4di 288	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦)))
206	146	eqeq2d 2749	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦)))
207	206	ralbidva 3119	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦)))
208	207	adantrl 712	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦)))
209	51	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
210	209	raleqdv 3339	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦)))
211	205, 208, 210	3bitr3rd 309	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦)))
212	123	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐾 → ((𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ (𝑐‘𝑦) = (𝑏‘𝑦)))
213		eqcom 2745	. . . . . . . 8 ⊢ ((𝑐‘𝑦) = (𝑏‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦))
214	212, 213	bitrdi 286	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 → ((𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦)))
215	214	ralbiia 3089	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦))
216	211, 215	bitr4di 288	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦)))
217	36	adantrr 713	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏 Fn (1...(𝑁 + 1)))
218	145	adantrl 712	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
219		f1ofn 6701	. . . . . . 7 ⊢ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) Fn (1...(𝑁 + 1)))
220	218, 219	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) Fn (1...(𝑁 + 1)))
221		eqfnfv 6891	. . . . . 6 ⊢ ((𝑏 Fn (1...(𝑁 + 1)) ∧ (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) Fn (1...(𝑁 + 1))) → (𝑏 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦)))
222	217, 220, 221	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦)))
223	143	adantrl 712	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐:𝐾–1-1-onto→𝐾)
224		f1ofn 6701	. . . . . . 7 ⊢ (𝑐:𝐾–1-1-onto→𝐾 → 𝑐 Fn 𝐾)
225	223, 224	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 Fn 𝐾)
226	115	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝐾 ⊆ (1...(𝑁 + 1)))
227	217, 226	fnssresd 6540	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 ↾ 𝐾) Fn 𝐾)
228		eqfnfv 6891	. . . . . 6 ⊢ ((𝑐 Fn 𝐾 ∧ (𝑏 ↾ 𝐾) Fn 𝐾) → (𝑐 = (𝑏 ↾ 𝐾) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦)))
229	225, 227, 228	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 = (𝑏 ↾ 𝐾) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦)))
230	216, 222, 229	3bitr4d 310	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ↔ 𝑐 = (𝑏 ↾ 𝐾)))
231	12, 130, 190, 230	f1o2d 7501	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾)):𝐵–1-1-onto→𝐶)
232	11, 231	hasheqf1od 13996	. 2 ⊢ (𝜑 → (♯‘𝐵) = (♯‘𝐶))
233	128	fveq2i 6759	. . . 4 ⊢ (♯‘𝐶) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)})
234		fzfi 13620	. . . . . . 7 ⊢ (2...(𝑁 + 1)) ∈ Fin
235		diffi 8979	. . . . . . 7 ⊢ ((2...(𝑁 + 1)) ∈ Fin → ((2...(𝑁 + 1)) ∖ {𝑀}) ∈ Fin)
236	234, 235	ax-mp 5	. . . . . 6 ⊢ ((2...(𝑁 + 1)) ∖ {𝑀}) ∈ Fin
237	49, 236	eqeltri 2835	. . . . 5 ⊢ 𝐾 ∈ Fin
238	44	derangval 33029	. . . . 5 ⊢ (𝐾 ∈ Fin → (𝐷‘𝐾) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}))
239	237, 238	ax-mp 5	. . . 4 ⊢ (𝐷‘𝐾) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)})
240	44, 45	derangen2 33036	. . . . 5 ⊢ (𝐾 ∈ Fin → (𝐷‘𝐾) = (𝑆‘(♯‘𝐾)))
241	237, 240	ax-mp 5	. . . 4 ⊢ (𝐷‘𝐾) = (𝑆‘(♯‘𝐾))
242	233, 239, 241	3eqtr2ri 2773	. . 3 ⊢ (𝑆‘(♯‘𝐾)) = (♯‘𝐶)
243	50	simp3d 1142	. . . 4 ⊢ (𝜑 → (♯‘𝐾) = (𝑁 − 1))
244	243	fveq2d 6760	. . 3 ⊢ (𝜑 → (𝑆‘(♯‘𝐾)) = (𝑆‘(𝑁 − 1)))
245	242, 244	eqtr3id 2793	. 2 ⊢ (𝜑 → (♯‘𝐶) = (𝑆‘(𝑁 − 1)))
246	232, 245	eqtrd 2778	1 ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {cab 2715 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 {cpr 4560 ⟨cop 4564 ↦ cmpt 5153 ^◡ccnv 5579 ↾ cres 5582 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 –1-1→wf1 6415 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 1c1 10803 + caddc 10805 − cmin 11135 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤ_≥cuz 12511 ...cfz 13168 ♯chash 13972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-fz 13169 df-hash 13973

This theorem is referenced by: subfacp1lem6 33047

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