subfacp1lem3 - Mathbox for Mario Carneiro

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Theorem subfacp1lem3 35187

Description: Lemma for subfacp1 35191. In subfacp1lem6 35190 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 14013	. . . . . . . 8 ⊢ (1...(𝑁 + 1)) ∈ Fin
3		deranglem 35171	. . . . . . . 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 2837	. . . . . 6 ⊢ 𝐴 ∈ Fin
6		subfacp1lem3.b	. . . . . . 7 ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)}
7	6	ssrab3 4082	. . . . . 6 ⊢ 𝐵 ⊆ 𝐴
8		ssfi 9213	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
9	5, 7, 8	mp2an 692	. . . . 5 ⊢ 𝐵 ∈ Fin
10	9	elexi 3503	. . . 4 ⊢ 𝐵 ∈ V
11	10	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
12		eqid 2737	. . . 4 ⊢ (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾)) = (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾))
13		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
14		fveq1 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑏 → (𝑔‘1) = (𝑏‘1))
15	14	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑏 → ((𝑔‘1) = 𝑀 ↔ (𝑏‘1) = 𝑀))
16		fveq1 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑏 → (𝑔‘𝑀) = (𝑏‘𝑀))
17	16	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑏 → ((𝑔‘𝑀) = 1 ↔ (𝑏‘𝑀) = 1))
18	15, 17	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑏 → (((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1) ↔ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1)))
19	18, 6	elrab2 3695	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1)))
20	13, 19	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1)))
21	20	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐴)
22		vex 3484	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
23		f1oeq1 6836	. . . . . . . . . . . 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 6905	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑏 → (𝑓‘𝑦) = (𝑏‘𝑦))
25	24	neeq1d 3000	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑏 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦))
26	25	ralbidv 3178	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))
27	23, 26	anbi12d 632	. . . . . . . . . . 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 3682	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐴 ↔ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))
29	21, 28	sylib 218	. . . . . . . . 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 6847	. . . . . . . 8 ⊢ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)))
32		df-f1 6566	. . . . . . . . 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 6849	. . . . . . . . . . 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 6687	. . . . . . . . . 10 ⊢ (𝑏 Fn (1...(𝑁 + 1)) → (𝑏 ↾ (1...(𝑁 + 1))) = 𝑏)
38		f1oeq1 6836	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
41		f1ofo 6855	. . . . . . . 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 4179	. . . . . . . . . . . 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 35184	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))
51	50	simp2d 1144	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
52	43, 51	sseqtrid 4026	. . . . . . . . . . 11 ⊢ (𝜑 → {1, 𝑀} ⊆ (1...(𝑁 + 1)))
53	52	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → {1, 𝑀} ⊆ (1...(𝑁 + 1)))
54	36, 53	fnssresd 6692	. . . . . . . . 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 4763	. . . . . . . . . . . 12 ⊢ 𝑀 ∈ {1, 𝑀}
58	56, 57	eqeltrdi 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘1) ∈ {1, 𝑀})
59	55	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) = 1)
60		1ex 11257	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
61	60	prid1 4762	. . . . . . . . . . . 12 ⊢ 1 ∈ {1, 𝑀}
62	59, 61	eqeltrdi 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) ∈ {1, 𝑀})
63		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝑏‘𝑥) = (𝑏‘1))
64	63	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((𝑏‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘1) ∈ {1, 𝑀}))
65		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑀 → (𝑏‘𝑥) = (𝑏‘𝑀))
66	65	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑀 → ((𝑏‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘𝑀) ∈ {1, 𝑀}))
67	60, 48, 64, 66	ralpr 4700	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀} ↔ ((𝑏‘1) ∈ {1, 𝑀} ∧ (𝑏‘𝑀) ∈ {1, 𝑀}))
68	58, 62, 67	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀})
69		fvres 6925	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {1, 𝑀} → ((𝑏 ↾ {1, 𝑀})‘𝑥) = (𝑏‘𝑥))
70	69	eleq1d 2826	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {1, 𝑀} → (((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘𝑥) ∈ {1, 𝑀}))
71	70	ralbiia 3091	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀} ↔ ∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀})
72	68, 71	sylibr 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀})
73		ffnfv 7139	. . . . . . . . 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 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀})
75		fveqeq2 6915	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) = 1 ↔ (𝑏‘𝑀) = 1))
76	75	rspcev 3622	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ {1, 𝑀} ∧ (𝑏‘𝑀) = 1) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1)
77	57, 59, 76	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1)
78		fveqeq2 6915	. . . . . . . . . . . 12 ⊢ (𝑦 = 1 → ((𝑏‘𝑦) = 𝑀 ↔ (𝑏‘1) = 𝑀))
79	78	rspcev 3622	. . . . . . . . . . 11 ⊢ ((1 ∈ {1, 𝑀} ∧ (𝑏‘1) = 𝑀) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀)
80	61, 56, 79	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀)
81		eqeq2 2749	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((𝑏‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 1))
82	81	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1))
83		eqeq2 2749	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑀 → ((𝑏‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 𝑀))
84	83	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀))
85	60, 48, 82, 84	ralpr 4700	. . . . . . . . . 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 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥)
87		eqcom 2744	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ((𝑏 ↾ {1, 𝑀})‘𝑦) = 𝑥)
88		fvres 6925	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {1, 𝑀} → ((𝑏 ↾ {1, 𝑀})‘𝑦) = (𝑏‘𝑦))
89	88	eqeq1d 2739	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {1, 𝑀} → (((𝑏 ↾ {1, 𝑀})‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 𝑥))
90	87, 89	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {1, 𝑀} → (𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ (𝑏‘𝑦) = 𝑥))
91	90	rexbiia 3092	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥)
92	91	ralbii 3093	. . . . . . . . 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 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦))
94		dffo3 7122	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}):{1, 𝑀}–onto→{1, 𝑀})
96		resdif 6869	. . . . . . 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 1373	. . . . . 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 4158	. . . . . . . . . 10 ⊢ ({1, 𝑀} ∪ 𝐾) = (𝐾 ∪ {1, 𝑀})
99	98, 51	eqtrid 2789	. . . . . . . . 9 ⊢ (𝜑 → ({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)))
100		incom 4209	. . . . . . . . . . 11 ⊢ ({1, 𝑀} ∩ 𝐾) = (𝐾 ∩ {1, 𝑀})
101	50	simp1d 1143	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∩ {1, 𝑀}) = ∅)
102	100, 101	eqtrid 2789	. . . . . . . . . 10 ⊢ (𝜑 → ({1, 𝑀} ∩ 𝐾) = ∅)
103		uneqdifeq 4493	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ (𝜑 → (({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾))
105	99, 104	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾)
106	105	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾)
107		reseq2 5992	. . . . . . . . 9 ⊢ (((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾 → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})) = (𝑏 ↾ 𝐾))
108	107	f1oeq1d 6843	. . . . . . . 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 6837	. . . . . . . 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 6838	. . . . . . . 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 305	. . . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾)
114		ssun1 4178	. . . . . . . 8 ⊢ 𝐾 ⊆ (𝐾 ∪ {1, 𝑀})
115	114, 51	sseqtrid 4026	. . . . . . 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 4052	. . . . . 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 6047	. . . . . 6 ⊢ (𝑏 ↾ 𝐾) ∈ V
121		f1oeq1 6836	. . . . . . 7 ⊢ (𝑓 = (𝑏 ↾ 𝐾) → (𝑓:𝐾–1-1-onto→𝐾 ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾))
122		fveq1 6905	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏 ↾ 𝐾) → (𝑓‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦))
123		fvres 6925	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐾 → ((𝑏 ↾ 𝐾)‘𝑦) = (𝑏‘𝑦))
124	122, 123	sylan9eq 2797	. . . . . . . . 9 ⊢ ((𝑓 = (𝑏 ↾ 𝐾) ∧ 𝑦 ∈ 𝐾) → (𝑓‘𝑦) = (𝑏‘𝑦))
125	124	neeq1d 3000	. . . . . . . 8 ⊢ ((𝑓 = (𝑏 ↾ 𝐾) ∧ 𝑦 ∈ 𝐾) → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦))
126	125	ralbidva 3176	. . . . . . 7 ⊢ (𝑓 = (𝑏 ↾ 𝐾) → (∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦))
127	121, 126	anbi12d 632	. . . . . 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 3682	. . . . 5 ⊢ ((𝑏 ↾ 𝐾) ∈ 𝐶 ↔ ((𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦))
130	113, 119, 129	sylanbrc 583	. . . 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 2737	. . . . . . . 8 ⊢ (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})
134		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ 𝐶)
135		vex 3484	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
136		f1oeq1 6836	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑐 → (𝑓:𝐾–1-1-onto→𝐾 ↔ 𝑐:𝐾–1-1-onto→𝐾))
137		fveq1 6905	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑐 → (𝑓‘𝑦) = (𝑐‘𝑦))
138	137	neeq1d 3000	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑐 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑐‘𝑦) ≠ 𝑦))
139	138	ralbidv 3178	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑐 → (∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦))
140	136, 139	anbi12d 632	. . . . . . . . . . 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 3682	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐶 ↔ (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦))
142	134, 141	sylib 218	. . . . . . . . 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 35185	. . . . . . 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 1143	. . . . . 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 35186	. . . . . . . . . 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 3251	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → (𝑐‘𝑦) ≠ 𝑦)
149	146, 148	eqnetrd 3008	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
150	149	ralrimiva 3146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ 𝐾 ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
151	144	simp2d 1144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀)
152		elfzuz 13560	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (2...(𝑁 + 1)) → 𝑀 ∈ (ℤ_≥‘2))
153		eluz2b3 12964	. . . . . . . . . . . . 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 3008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) ≠ 1)
158	144	simp3d 1145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1)
159	156	necomd 2996	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 1 ≠ 𝑀)
160	158, 159	eqnetrd 3008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) ≠ 𝑀)
161		fveq2 6906	. . . . . . . . . . 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 3002	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) ≠ 1))
164		fveq2 6906	. . . . . . . . . . 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 3002	. . . . . . . . . 10 ⊢ (𝑦 = 𝑀 → (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) ≠ 𝑀))
167	60, 48, 163, 166	ralpr 4700	. . . . . . . . 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 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ {1, 𝑀} ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
169		ralunb 4197	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
171	51	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
172	170, 171	raleqtrdv 3328	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦)
173		prex 5437	. . . . . . . 8 ⊢ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩} ∈ V
174	135, 173	unex 7764	. . . . . . 7 ⊢ (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ V
175		f1oeq1 6836	. . . . . . . 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))))
176		fveq1 6905	. . . . . . . . . 10 ⊢ (𝑓 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (𝑓‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦))
177	176	neeq1d 3000	. . . . . . . . 9 ⊢ (𝑓 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → ((𝑓‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ≠ 𝑦))
178	177	ralbidv 3178	. . . . . . . 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))
179	175, 178	anbi12d 632	. . . . . . 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)))
180	174, 179, 1	elab2 3682	. . . . . 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))
181	145, 172, 180	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ 𝐴)
182	151, 158	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀 ∧ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1))
183		fveq1 6905	. . . . . . . 8 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (𝑔‘1) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1))
184	183	eqeq1d 2739	. . . . . . 7 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → ((𝑔‘1) = 𝑀 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀))
185		fveq1 6905	. . . . . . . 8 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → (𝑔‘𝑀) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀))
186	185	eqeq1d 2739	. . . . . . 7 ⊢ (𝑔 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) → ((𝑔‘𝑀) = 1 ↔ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1))
187	184, 186	anbi12d 632	. . . . . 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)))
188	187, 6	elrab2 3695	. . . . 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)))
189	181, 182, 188	sylanbrc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ∈ 𝐵)
190	56	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘1) = 𝑀)
191	151	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1) = 𝑀)
192	190, 191	eqtr4d 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘1) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1))
193	59	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘𝑀) = 1)
194	158	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀) = 1)
195	193, 194	eqtr4d 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘𝑀) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀))
196		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑦 = 1 → (𝑏‘𝑦) = (𝑏‘1))
197	196, 161	eqeq12d 2753	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → ((𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ (𝑏‘1) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘1)))
198		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → (𝑏‘𝑦) = (𝑏‘𝑀))
199	198, 164	eqeq12d 2753	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ (𝑏‘𝑀) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑀)))
200	60, 48, 197, 199	ralpr 4700	. . . . . . . . . 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)))
201	192, 195, 200	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦))
202	201	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))))
203		ralunb 4197	. . . . . . . 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)))
204	202, 203	bitr4di 289	. . . . . . 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)))
205	146	eqeq2d 2748	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦)))
206	205	ralbidva 3176	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦)))
207	206	adantrl 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦)))
208	51	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
209	208	raleqdv 3326	. . . . . . 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)))
210	204, 207, 209	3bitr3rd 310	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦)))
211	123	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐾 → ((𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ (𝑐‘𝑦) = (𝑏‘𝑦)))
212		eqcom 2744	. . . . . . . 8 ⊢ ((𝑐‘𝑦) = (𝑏‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦))
213	211, 212	bitrdi 287	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 → ((𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦)))
214	213	ralbiia 3091	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦))
215	210, 214	bitr4di 289	. . . . 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)))
216	36	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏 Fn (1...(𝑁 + 1)))
217	145	adantrl 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
218		f1ofn 6849	. . . . . . 7 ⊢ ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) Fn (1...(𝑁 + 1)))
219	217, 218	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) Fn (1...(𝑁 + 1)))
220		eqfnfv 7051	. . . . . 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)))
221	216, 219, 220	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})‘𝑦)))
222	143	adantrl 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐:𝐾–1-1-onto→𝐾)
223		f1ofn 6849	. . . . . . 7 ⊢ (𝑐:𝐾–1-1-onto→𝐾 → 𝑐 Fn 𝐾)
224	222, 223	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 Fn 𝐾)
225	115	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝐾 ⊆ (1...(𝑁 + 1)))
226	216, 225	fnssresd 6692	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 ↾ 𝐾) Fn 𝐾)
227		eqfnfv 7051	. . . . . 6 ⊢ ((𝑐 Fn 𝐾 ∧ (𝑏 ↾ 𝐾) Fn 𝐾) → (𝑐 = (𝑏 ↾ 𝐾) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦)))
228	224, 226, 227	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 = (𝑏 ↾ 𝐾) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦)))
229	215, 221, 228	3bitr4d 311	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝑐 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩}) ↔ 𝑐 = (𝑏 ↾ 𝐾)))
230	12, 130, 189, 229	f1o2d 7687	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾)):𝐵–1-1-onto→𝐶)
231	11, 230	hasheqf1od 14392	. 2 ⊢ (𝜑 → (♯‘𝐵) = (♯‘𝐶))
232	128	fveq2i 6909	. . . 4 ⊢ (♯‘𝐶) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)})
233		fzfi 14013	. . . . . . 7 ⊢ (2...(𝑁 + 1)) ∈ Fin
234		diffi 9215	. . . . . . 7 ⊢ ((2...(𝑁 + 1)) ∈ Fin → ((2...(𝑁 + 1)) ∖ {𝑀}) ∈ Fin)
235	233, 234	ax-mp 5	. . . . . 6 ⊢ ((2...(𝑁 + 1)) ∖ {𝑀}) ∈ Fin
236	49, 235	eqeltri 2837	. . . . 5 ⊢ 𝐾 ∈ Fin
237	44	derangval 35172	. . . . 5 ⊢ (𝐾 ∈ Fin → (𝐷‘𝐾) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}))
238	236, 237	ax-mp 5	. . . 4 ⊢ (𝐷‘𝐾) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)})
239	44, 45	derangen2 35179	. . . . 5 ⊢ (𝐾 ∈ Fin → (𝐷‘𝐾) = (𝑆‘(♯‘𝐾)))
240	236, 239	ax-mp 5	. . . 4 ⊢ (𝐷‘𝐾) = (𝑆‘(♯‘𝐾))
241	232, 238, 240	3eqtr2ri 2772	. . 3 ⊢ (𝑆‘(♯‘𝐾)) = (♯‘𝐶)
242	50	simp3d 1145	. . . 4 ⊢ (𝜑 → (♯‘𝐾) = (𝑁 − 1))
243	242	fveq2d 6910	. . 3 ⊢ (𝜑 → (𝑆‘(♯‘𝐾)) = (𝑆‘(𝑁 − 1)))
244	241, 243	eqtr3id 2791	. 2 ⊢ (𝜑 → (♯‘𝐶) = (𝑆‘(𝑁 − 1)))
245	231, 244	eqtrd 2777	1 ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 {cpr 4628 ⟨cop 4632 ↦ cmpt 5225 ^◡ccnv 5684 ↾ cres 5687 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 –1-1→wf1 6558 –onto→wfo 6559 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 1c1 11156 + caddc 11158 − cmin 11492 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤ_≥cuz 12878 ...cfz 13547 ♯chash 14369

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370

This theorem is referenced by: subfacp1lem6 35190

W3C validator