Theorem ixpfi2 9362

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 4213	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
3		ssfi 9187	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (𝐴 ∩ 𝐶) ⊆ 𝐶) → (𝐴 ∩ 𝐶) ∈ Fin)
4	1, 2, 3	sylancl 586	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ∈ Fin)
5		inss1 4212	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴
6		ixpfi2.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
7	6	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin)
8		ssralv 4027	. . . 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 9361	. . 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 8947	. . . . 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 3463	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
17	16	elixp 8918	. . . . . . . . . 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 3463	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
22	21	elixp 8918	. . . . . . . . . 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 3098	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
26		difss 4111	. . . . . . . . . . 11 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
27		ssralv 4027	. . . . . . . . . . 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 3957	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ {𝐷}))
31		elsni 4618	. . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑔‘𝑥) ∈ 𝐵 → (𝑔‘𝑥) ∈ {𝐷}))
34		elsni 4618	. . . . . . . . . . . . . . 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 2757	. . . . . . . . . . . . 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 3152	. . . . . . . . . . 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 6895	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑓‘𝑥))
46		fvres 6895	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑔‘𝑥))
47	45, 46	eqeq12d 2751	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → (((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ (𝑓‘𝑥) = (𝑔‘𝑥)))
48	47	ralbiia 3080	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐶)((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))
49		inundif 4454	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) = 𝐴
50	49	raleqi 3303	. . . . . . 7 ⊢ (∀𝑥 ∈ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶))(𝑓‘𝑥) = (𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥))
51		ralunb 4172	. . . . . . 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 6661	. . . . . . 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 6661	. . . . . . 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 7021	. . . . . 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 7021	. . . . . 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 9006	. 2 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↦ (𝑓 ↾ (𝐴 ∩ 𝐶))):X𝑥 ∈ 𝐴 𝐵–1-1→X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
67		f1fi 9324	. 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 2108  ∀wral 3051   ∖ cdif 3923   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926  {csn 4601   ↦ cmpt 5201   ↾ cres 5656   Fn wfn 6526  –1-1→wf1 6528  ‘cfv 6531  Xcixp 8911  Fincfn 8959

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-1o 8480  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-fin 8963

This theorem is referenced by:  psrbaglefi  21886  eulerpartlemb  34400