Theorem ixpfi2 9240

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 4189	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
3		ssfi 9087	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (𝐴 ∩ 𝐶) ⊆ 𝐶) → (𝐴 ∩ 𝐶) ∈ Fin)
4	1, 2, 3	sylancl 586	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ∈ Fin)
5		inss1 4188	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴
6		ixpfi2.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
7	6	ralrimiva 3121	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin)
8		ssralv 4004	. . . 4 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ Fin → ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin))
9	5, 7, 8	mpsyl 68	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
10		ixpfi 9239	. . 3 ⊢ (((𝐴 ∩ 𝐶) ∈ Fin ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin) → X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
11	4, 9, 10	syl2anc 584	. 2 ⊢ (𝜑 → X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
12		resixp 8860	. . . . 5 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → (𝑓 ↾ (𝐴 ∩ 𝐶)) ∈ X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
13	5, 12	mpan 690	. . . 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 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
16		vex 3440	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
17	16	elixp 8831	. . . . . . . . . 10 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
18	15, 17	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
19	18	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)
20		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)
21		vex 3440	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
22	21	elixp 8831	. . . . . . . . . 10 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
23	20, 22	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
24	23	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
25		r19.26 3089	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
26		difss 4087	. . . . . . . . . . 11 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
27		ssralv 4004	. . . . . . . . . . 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 3934	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ {𝐷}))
31		elsni 4594	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥) ∈ {𝐷} → (𝑓‘𝑥) = 𝐷)
32	30, 31	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) = 𝐷))
33	29	sseld 3934	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑔‘𝑥) ∈ 𝐵 → (𝑔‘𝑥) ∈ {𝐷}))
34		elsni 4594	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘𝑥) ∈ {𝐷} → (𝑔‘𝑥) = 𝐷)
35	33, 34	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑔‘𝑥) ∈ 𝐵 → (𝑔‘𝑥) = 𝐷))
36	32, 35	anim12d 609	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ((𝑓‘𝑥) = 𝐷 ∧ (𝑔‘𝑥) = 𝐷)))
37		eqtr3 2751	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑥) = 𝐷 ∧ (𝑔‘𝑥) = 𝐷) → (𝑓‘𝑥) = (𝑔‘𝑥))
38	36, 37	syl6 35	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → (𝑓‘𝑥) = (𝑔‘𝑥)))
39	38	ralimdva 3141	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∖ 𝐶)((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥)))
40	39	adantr 480	. . . . . . . . . 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 34	. . . . . . . . 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 243	. . . . . . . 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 699	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))
44	43	biantrud 531	. . . . . 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 6841	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑓‘𝑥))
46		fvres 6841	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑔‘𝑥))
47	45, 46	eqeq12d 2745	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → (((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ (𝑓‘𝑥) = (𝑔‘𝑥)))
48	47	ralbiia 3073	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐶)((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))
49		inundif 4430	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) = 𝐴
50	49	raleqi 3287	. . . . . . 7 ⊢ (∀𝑥 ∈ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶))(𝑓‘𝑥) = (𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥))
51		ralunb 4148	. . . . . . 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 277	. . . . . 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 314	. . . . 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 494	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑓 Fn 𝐴)
55		fnssres 6605	. . . . . . 7 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴) → (𝑓 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
56	54, 5, 55	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑓 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
57	23	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑔 Fn 𝐴)
58		fnssres 6605	. . . . . . 7 ⊢ ((𝑔 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴) → (𝑔 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
59	57, 5, 58	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑔 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
60		eqfnfv 6965	. . . . . 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 584	. . . . 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 6965	. . . . . 6 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑔 Fn 𝐴) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥)))
63	54, 57, 62	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥)))
64	53, 61, 63	3bitr4d 311	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ((𝑓 ↾ (𝐴 ∩ 𝐶)) = (𝑔 ↾ (𝐴 ∩ 𝐶)) ↔ 𝑓 = 𝑔))
65	64	ex 412	. . 3 ⊢ (𝜑 → ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵) → ((𝑓 ↾ (𝐴 ∩ 𝐶)) = (𝑔 ↾ (𝐴 ∩ 𝐶)) ↔ 𝑓 = 𝑔)))
66	14, 65	dom2lem 8917	. 2 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↦ (𝑓 ↾ (𝐴 ∩ 𝐶))):X𝑥 ∈ 𝐴 𝐵–1-1→X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
67		f1fi 9203	. 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 584	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  {csn 4577   ↦ cmpt 5173   ↾ cres 5621   Fn wfn 6477  –1-1→wf1 6479  ‘cfv 6482  Xcixp 8824  Fincfn 8872

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-1o 8388  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-fin 8876

This theorem is referenced by:  psrbaglefi  21833  eulerpartlemb  34336