Theorem ixpfi2 9248

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 4188	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
3		ssfi 9095	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (𝐴 ∩ 𝐶) ⊆ 𝐶) → (𝐴 ∩ 𝐶) ∈ Fin)
4	1, 2, 3	sylancl 586	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ∈ Fin)
5		inss1 4187	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴
6		ixpfi2.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
7	6	ralrimiva 3126	. . . 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 68	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
10		ixpfi 9247	. . 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 8869	. . . . 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 3442	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
17	16	elixp 8840	. . . . . . . . . 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 3442	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
22	21	elixp 8840	. . . . . . . . . 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 3094	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
26		difss 4086	. . . . . . . . . . 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 4595	. . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑔‘𝑥) ∈ 𝐵 → (𝑔‘𝑥) ∈ {𝐷}))
34		elsni 4595	. . . . . . . . . . . . . . 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 2756	. . . . . . . . . . . . 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 3146	. . . . . . . . . . 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 6851	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑓‘𝑥))
46		fvres 6851	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑔‘𝑥))
47	45, 46	eqeq12d 2750	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → (((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ (𝑓‘𝑥) = (𝑔‘𝑥)))
48	47	ralbiia 3078	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐶)((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))
49		inundif 4429	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) = 𝐴
50	49	raleqi 3292	. . . . . . 7 ⊢ (∀𝑥 ∈ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶))(𝑓‘𝑥) = (𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥))
51		ralunb 4147	. . . . . . 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 6613	. . . . . . 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 6613	. . . . . . 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 6974	. . . . . 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 6974	. . . . . 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 8927	. 2 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↦ (𝑓 ↾ (𝐴 ∩ 𝐶))):X𝑥 ∈ 𝐴 𝐵–1-1→X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
67		f1fi 9212	. 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 1541   ∈ wcel 2113  ∀wral 3049   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  {csn 4578   ↦ cmpt 5177   ↾ cres 5624   Fn wfn 6485  –1-1→wf1 6487  ‘cfv 6490  Xcixp 8833  Fincfn 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-1o 8395  df-map 8763  df-pm 8764  df-ixp 8834  df-en 8882  df-dom 8883  df-fin 8885

This theorem is referenced by:  psrbaglefi  21880  eulerpartlemb  34474