Theorem ixpfi2 9161

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 4169	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
3		ssfi 8994	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (𝐴 ∩ 𝐶) ⊆ 𝐶) → (𝐴 ∩ 𝐶) ∈ Fin)
4	1, 2, 3	sylancl 587	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ∈ Fin)
5		inss1 4168	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴
6		ixpfi2.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
7	6	ralrimiva 3140	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin)
8		ssralv 3992	. . . 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 9160	. . 3 ⊢ (((𝐴 ∩ 𝐶) ∈ Fin ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin) → X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
11	4, 9, 10	syl2anc 585	. 2 ⊢ (𝜑 → X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ Fin)
12		resixp 8752	. . . . 5 ⊢ (((𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → (𝑓 ↾ (𝐴 ∩ 𝐶)) ∈ X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
13	5, 12	mpan 688	. . . 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 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵)
16		vex 3441	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
17	16	elixp 8723	. . . . . . . . . 10 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
18	15, 17	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
19	18	simprd 497	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)
20		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)
21		vex 3441	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
22	21	elixp 8723	. . . . . . . . . 10 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
23	20, 22	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
24	23	simprd 497	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
25		r19.26 3111	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
26		difss 4072	. . . . . . . . . . 11 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
27		ssralv 3992	. . . . . . . . . . 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 3925	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ {𝐷}))
31		elsni 4582	. . . . . . . . . . . . . . 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 3925	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑔‘𝑥) ∈ 𝐵 → (𝑔‘𝑥) ∈ {𝐷}))
34		elsni 4582	. . . . . . . . . . . . . . 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 610	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ((𝑓‘𝑥) = 𝐷 ∧ (𝑔‘𝑥) = 𝐷)))
37		eqtr3 2762	. . . . . . . . . . . . 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 3161	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∖ 𝐶)((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑔‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥)))
40	39	adantr 482	. . . . . . . . . 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	syl5bir 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 697	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ (𝐴 ∖ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))
44	43	biantrud 533	. . . . . 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 6823	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑓‘𝑥))
46		fvres 6823	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) = (𝑔‘𝑥))
47	45, 46	eqeq12d 2752	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → (((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ (𝑓‘𝑥) = (𝑔‘𝑥)))
48	47	ralbiia 3091	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐶)((𝑓 ↾ (𝐴 ∩ 𝐶))‘𝑥) = ((𝑔 ↾ (𝐴 ∩ 𝐶))‘𝑥) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)(𝑓‘𝑥) = (𝑔‘𝑥))
49		inundif 4418	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) = 𝐴
50	49	raleqi 3358	. . . . . . 7 ⊢ (∀𝑥 ∈ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶))(𝑓‘𝑥) = (𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥))
51		ralunb 4131	. . . . . . 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 496	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑓 Fn 𝐴)
55		fnssres 6586	. . . . . . 7 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴) → (𝑓 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
56	54, 5, 55	sylancl 587	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑓 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
57	23	simpld 496	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → 𝑔 Fn 𝐴)
58		fnssres 6586	. . . . . . 7 ⊢ ((𝑔 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴) → (𝑔 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
59	57, 5, 58	sylancl 587	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)) → (𝑔 ↾ (𝐴 ∩ 𝐶)) Fn (𝐴 ∩ 𝐶))
60		eqfnfv 6941	. . . . . 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 585	. . . . 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 6941	. . . . . 6 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑔 Fn 𝐴) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = (𝑔‘𝑥)))
63	54, 57, 62	syl2anc 585	. . . . 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 414	. . 3 ⊢ (𝜑 → ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵) → ((𝑓 ↾ (𝐴 ∩ 𝐶)) = (𝑔 ↾ (𝐴 ∩ 𝐶)) ↔ 𝑓 = 𝑔)))
66	14, 65	dom2lem 8813	. 2 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↦ (𝑓 ↾ (𝐴 ∩ 𝐶))):X𝑥 ∈ 𝐴 𝐵–1-1→X𝑥 ∈ (𝐴 ∩ 𝐶)𝐵)
67		f1fi 9150	. 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 585	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062   ∖ cdif 3889   ∪ cun 3890   ∩ cin 3891   ⊆ wss 3892  {csn 4565   ↦ cmpt 5164   ↾ cres 5602   Fn wfn 6453  –1-1→wf1 6455  ‘cfv 6458  Xcixp 8716  Fincfn 8764

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-1o 8328  df-er 8529  df-map 8648  df-pm 8649  df-ixp 8717  df-en 8765  df-dom 8766  df-fin 8768

This theorem is referenced by:  psrbaglefi  21180  psrbaglefiOLD  21181  eulerpartlemb  32380