Theorem ixpfi2 9318

Description: A Cartesian product of finite sets such that all but finitely many are singletons is finite. (Note that 𝐵(𝑥) and 𝐷(𝑥) are both possibly dependent on 𝑥.) (Contributed by Mario Carneiro, 25-Jan-2015.)

Hypotheses

Ref	Expression
ixpfi2.1	⊢ (𝜑 → 𝐶 ∈ Fin)
ixpfi2.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
ixpfi2.3	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝐵 ⊆ {𝐷})

Assertion

Ref	Expression
ixpfi2	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem ixpfi2

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixpfi2.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Fin)
2		inss2 4183	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
3		ssfi 9168	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (𝐴 ∩ 𝐶) ⊆ 𝐶) → (𝐴 ∩ 𝐶) ∈ Fin)
4	1, 2, 3	sylancl 598	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ∈ Fin)
5		inss1 4182	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴
6		ixpfi2.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
7	6	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin)
8		ssralv 4000	. . . 4 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ Fin → ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin))
9	5, 7, 8	mpsyl 69	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
10		ixpfi 9317	. . 3 ⊢ (((𝐴 ∩ 𝐶) ∈ Fin ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin) → X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
11	4, 9, 10	syl2anc 596	. 2 ⊢ (𝜑 → X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
12		resixp 8941	. . . . 5 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → (𝑓 ↾ (𝐴 ∩ 𝐶)) ∈ X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
13	5, 12	mpan 703	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑓 ↾ (𝐴 ∩ 𝐶)) ∈ X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
14	13	a1i 11	. . 3 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑓 ↾ (𝐴 ∩ 𝐶)) ∈ X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵))
15		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
16		vex 3454	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
17	16	elixp 8912	. . . . . . . . . 10 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
18	15, 17	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
19	18	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)
20		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)
21		vex 3454	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
22	21	elixp 8912	. . . . . . . . . 10 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
23	20, 22	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
24	23	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
25		r19.26 3122	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
26		difss 4083	. . . . . . . . . . 11 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
27		ssralv 4000	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ 𝐶) ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵)))
28	26, 27	ax-mp 5	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵))
29		ixpfi2.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝐵 ⊆ {𝐷})
30	29	sseld 3930	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ {𝐷}))
31		elsni 4601	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥) ∈ {𝐷} → (𝑓‘𝑥) = 𝐷)
32	30, 31	syl6 36	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) = 𝐷))
33	29	sseld 3930	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑔‘𝑥) ∈ 𝐵 → (𝑔‘𝑥) ∈ {𝐷}))
34		elsni 4601	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘𝑥) ∈ {𝐷} → (𝑔‘𝑥) = 𝐷)
35	33, 34	syl6 36	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑔‘𝑥) ∈ 𝐵 → (𝑔‘𝑥) = 𝐷))
36	32, 35	anim12d 621	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ((𝑓‘𝑥) = 𝐷 ∧ (𝑔‘𝑥) = 𝐷)))
37		eqtr3 2782	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑥) = 𝐷 ∧ (𝑔‘𝑥) = 𝐷) → (𝑓‘𝑥) = (𝑔‘𝑥))
38	36, 37	syl6 36	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → (𝑓‘𝑥) = (𝑔‘𝑥)))
39	38	ralimdva 3174	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∖ 𝐶)((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥)))
40	39	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (∀𝑥 ∈ (𝐴 ∖ 𝐶)((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥)))
41	28, 40	syl5 35	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥)))
42	25, 41	biimtrrid 246	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥)))
43	19, 24, 42	mp2and 712	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))
44	43	biantrud 541	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥) ↔ (∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥) ∧ ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))))
45		fvres 6898	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑓‘𝑥))
46		fvres 6898	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑔‘𝑥))
47	45, 46	eqeq12d 2776	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → (((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ (𝑓‘𝑥) = (𝑔‘𝑥)))
48	47	ralbiia 3106	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐶)((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))
49		inundif 4435	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) = 𝐴
50	49	raleqi 3317	. . . . . . 7 ⊢ (∀𝑥 ∈ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶))(𝑓‘𝑥) = (𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥))
51		ralunb 4143	. . . . . . 7 ⊢ (∀𝑥 ∈ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶))(𝑓‘𝑥) = (𝑔‘𝑥) ↔ (∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥) ∧ ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥)))
52	50, 51	bitr3i 280	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥) ↔ (∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥) ∧ ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥)))
53	44, 48, 52	3bitr4g 317	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (∀𝑥 ∈ (𝐴 ∩ 𝐶)((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥)))
54	18	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑓 Fn 𝐴)
55		fnssres 6656	. . . . . . 7 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴) → (𝑓 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
56	54, 5, 55	sylancl 598	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑓 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
57	23	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑔 Fn 𝐴)
58		fnssres 6656	. . . . . . 7 ⊢ ((𝑔 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴) → (𝑔 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
59	57, 5, 58	sylancl 598	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑔 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
60		eqfnfv 7023	. . . . . 6 ⊢ (((𝑓 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶) ∧ (𝑔 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶)) → ((𝑓 ↾ (𝐴 ∩ 𝐶)) = (𝑔 ↾ (𝐴 ∩ 𝐶)) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥)))
61	56, 59, 60	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ((𝑓 ↾ (𝐴 ∩ 𝐶)) = (𝑔 ↾ (𝐴 ∩ 𝐶)) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥)))
62		eqfnfv 7023	. . . . . 6 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑔 Fn 𝐴) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥)))
63	54, 57, 62	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥)))
64	53, 61, 63	3bitr4d 314	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ((𝑓 ↾ (𝐴 ∩ 𝐶)) = (𝑔 ↾ (𝐴 ∩ 𝐶)) ↔ 𝑓 = 𝑔))
65	64	ex 418	. . 3 ⊢ (𝜑 → ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵) → ((𝑓 ↾ (𝐴 ∩ 𝐶)) = (𝑔 ↾ (𝐴 ∩ 𝐶)) ↔ 𝑓 = 𝑔)))
66	14, 65	dom2lem 8999	. 2 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↦ (𝑓 ↾ (𝐴 ∩ 𝐶))):X𝑥 ∈ 𝐴 𝐵–1-1→X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
67		f1fi 9285	. 2 ⊢ ((X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↦ (𝑓 ↾ (𝐴 ∩ 𝐶))):X𝑥 ∈ 𝐴 𝐵–1-1→X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵) → X𝑥 ∈ 𝐴 𝐵 ∈ Fin)
68	11, 66, 67	syl2anc 596	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  {csn 4584   ↦ cmpt 5186   ↾ cres 5657   Fn wfn 6528  –1-1→wf1 6530  ‘cfv 6533  Xcixp 8905  Fincfn 8953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-1o 8456  df-map 8829  df-pm 8830  df-ixp 8906  df-en 8954  df-dom 8955  df-fin 8957

This theorem is used by:  psrbaglefi  22142  eulerpartlemb  34880