Theorem ptcld 23781

Description: A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Hypotheses

Ref	Expression
ptcld.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ptcld.f	⊢ (𝜑 → 𝐹:𝐴⟶Top)
ptcld.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘(𝐹‘𝑘)))

Assertion

Ref	Expression
ptcld	⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘𝐹)))

Distinct variable groups:   𝜑,𝑘   𝐴,𝑘   𝑘,𝐹   𝑘,𝑉

Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem ptcld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcld.c	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘(𝐹‘𝑘)))
2		eqid 2762	. . . . . 6 ⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘)
3	2	cldss 23197	. . . . 5 ⊢ (𝐶 ∈ (Clsd‘(𝐹‘𝑘)) → 𝐶 ⊆ ∪ (𝐹‘𝑘))
4	1, 3	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ⊆ ∪ (𝐹‘𝑘))
5	4	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘))
6		boxriin 8936	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘) → X𝑘 ∈ 𝐴 𝐶 = (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))))
7	5, 6	syl 18	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 = (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))))
8		ptcld.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9		ptcld.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶Top)
10		eqid 2762	. . . . . 6 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
11	10	ptuni 23762	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ (∏t‘𝐹))
12	8, 9, 11	syl2anc 595	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ (∏t‘𝐹))
13	12	ineq1d 4171	. . 3 ⊢ (𝜑 → (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = (∪ (∏t‘𝐹) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))))
14		pttop 23750	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
15	8, 9, 14	syl2anc 595	. . . 4 ⊢ (𝜑 → (∏t‘𝐹) ∈ Top)
16		sseq1 3961	. . . . . . . . . . 11 ⊢ (𝐶 = if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) → (𝐶 ⊆ ∪ (𝐹‘𝑘) ↔ if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘)))
17		sseq1 3961	. . . . . . . . . . 11 ⊢ (∪ (𝐹‘𝑘) = if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) → (∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘) ↔ if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘)))
18		simpl 487	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ ∪ (𝐹‘𝑘) ∧ 𝑘 = 𝑥) → 𝐶 ⊆ ∪ (𝐹‘𝑘))
19		ssidd 3959	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ ∪ (𝐹‘𝑘) ∧ ¬ 𝑘 = 𝑥) → ∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘))
20	16, 17, 18, 19	ifbothda 4525	. . . . . . . . . 10 ⊢ (𝐶 ⊆ ∪ (𝐹‘𝑘) → if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
21	20	ralimi 3101	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
22		ss2ixp 8906	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
23	5, 21, 22	3syl 19	. . . . . . . 8 ⊢ (𝜑 → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
24	23	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
25	12	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ (∏t‘𝐹))
26	24, 25	sseqtrd 3972	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (∏t‘𝐹))
27	12	eqcomd 2768	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (∏t‘𝐹) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
28	27	difeq1d 4079	. . . . . . . . 9 ⊢ (𝜑 → (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))))
29	28	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))))
30		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
31	5	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘))
32		boxcutc 8937	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘)) → (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)))
33	30, 31, 32	syl2anc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)))
34		ixpeq2 8907	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)))
35		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
36	35	unieqd 4884	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑥))
37		csbeq1a 3866	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → 𝐶 = ⦋𝑥 / 𝑘⦌𝐶)
38	36, 37	difeq12d 4081	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → (∪ (𝐹‘𝑘) ∖ 𝐶) = (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶))
39	38	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝑘 = 𝑥) → (∪ (𝐹‘𝑘) ∖ 𝐶) = (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶))
40	39	ifeq1da 4518	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)))
41	34, 40	mprg 3084	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘))
42	41	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑘) ∖ 𝐶), ∪ (𝐹‘𝑘)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)))
43	29, 33, 42	3eqtrd 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)))
44	8	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉)
45	9	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶Top)
46	1	ralrimiva 3156	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(𝐹‘𝑘)))
47		nfv 1943	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝐶 ∈ (Clsd‘(𝐹‘𝑘))
48		nfcsb1v 3876	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶
49	48	nfel1 2940	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥))
50		2fveq3 6886	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → (Clsd‘(𝐹‘𝑘)) = (Clsd‘(𝐹‘𝑥)))
51	37, 50	eleq12d 2856	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → (𝐶 ∈ (Clsd‘(𝐹‘𝑘)) ↔ ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥))))
52	47, 49, 51	cbvralw 3306	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(𝐹‘𝑘)) ↔ ∀𝑥 ∈ 𝐴 ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)))
53	46, 52	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)))
54	53	r19.21bi 3256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)))
55		eqid 2762	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑥) = ∪ (𝐹‘𝑥)
56	55	cldopn 23199	. . . . . . . . 9 ⊢ (⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)) → (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶) ∈ (𝐹‘𝑥))
57	54, 56	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶) ∈ (𝐹‘𝑥))
58	44, 45, 57	ptopn2 23752	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)) ∈ (∏t‘𝐹))
59	43, 58	eqeltrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹))
60		eqid 2762	. . . . . . . . 9 ⊢ ∪ (∏t‘𝐹) = ∪ (∏t‘𝐹)
61	60	iscld 23195	. . . . . . . 8 ⊢ ((∏t‘𝐹) ∈ Top → (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈ (Clsd‘(∏t‘𝐹)) ↔ (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (∏t‘𝐹) ∧ (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹))))
62	15, 61	syl 18	. . . . . . 7 ⊢ (𝜑 → (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈ (Clsd‘(∏t‘𝐹)) ↔ (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (∏t‘𝐹) ∧ (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹))))
63	62	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈ (Clsd‘(∏t‘𝐹)) ↔ (X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (∏t‘𝐹) ∧ (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹))))
64	26, 59, 63	mpbir2and 725	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈ (Clsd‘(∏t‘𝐹)))
65	64	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈ (Clsd‘(∏t‘𝐹)))
66	60	riincld 23212	. . . 4 ⊢ (((∏t‘𝐹) ∈ Top ∧ ∀𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈ (Clsd‘(∏t‘𝐹))) → (∪ (∏t‘𝐹) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (Clsd‘(∏t‘𝐹)))
67	15, 65, 66	syl2anc 595	. . 3 ⊢ (𝜑 → (∪ (∏t‘𝐹) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (Clsd‘(∏t‘𝐹)))
68	13, 67	eqeltrd 2862	. 2 ⊢ (𝜑 → (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (Clsd‘(∏t‘𝐹)))
69	7, 68	eqeltrd 2862	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘𝐹)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ⦋csb 3852   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ifcif 4486  ∪ cuni 4871  ∩ ciin 4956  ⟶wf 6532  ‘cfv 6536  Xcixp 8893  ∏_tcpt 17497  Topctop 23061  Clsdccld 23184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7861  df-1o 8451  df-2o 8452  df-ixp 8894  df-en 8942  df-fin 8945  df-fi 9369  df-topgen 17502  df-pt 17503  df-top 23062  df-bases 23114  df-cld 23187

This theorem is used by:  ptcldmpt  23782