Theorem ptcld 23559

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 2735	. . . . . 6 ⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘)
3	2	cldss 22975	. . . . 5 ⊢ (𝐶 ∈ (Clsd‘(𝐹‘𝑘)) → 𝐶 ⊆ ∪ (𝐹‘𝑘))
4	1, 3	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ⊆ ∪ (𝐹‘𝑘))
5	4	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘))
6		boxriin 8880	. . 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 17	. 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 2735	. . . . . 6 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
11	10	ptuni 23540	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ (∏t‘𝐹))
12	8, 9, 11	syl2anc 585	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ (∏t‘𝐹))
13	12	ineq1d 4170	. . 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 23528	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
15	8, 9, 14	syl2anc 585	. . . 4 ⊢ (𝜑 → (∏t‘𝐹) ∈ Top)
16		sseq1 3958	. . . . . . . . . . 11 ⊢ (𝐶 = if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) → (𝐶 ⊆ ∪ (𝐹‘𝑘) ↔ if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘)))
17		sseq1 3958	. . . . . . . . . . 11 ⊢ (∪ (𝐹‘𝑘) = if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) → (∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘) ↔ if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘)))
18		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ ∪ (𝐹‘𝑘) ∧ 𝑘 = 𝑥) → 𝐶 ⊆ ∪ (𝐹‘𝑘))
19		ssidd 3956	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ ∪ (𝐹‘𝑘) ∧ ¬ 𝑘 = 𝑥) → ∪ (𝐹‘𝑘) ⊆ ∪ (𝐹‘𝑘))
20	16, 17, 18, 19	ifbothda 4517	. . . . . . . . . 10 ⊢ (𝐶 ⊆ ∪ (𝐹‘𝑘) → if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
21	20	ralimi 3072	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (𝐹‘𝑘))
22		ss2ixp 8850	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝜑 → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
24	23	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
25	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ (∏t‘𝐹))
26	24, 25	sseqtrd 3969	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ⊆ ∪ (∏t‘𝐹))
27	12	eqcomd 2741	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (∏t‘𝐹) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
28	27	difeq1d 4076	. . . . . . . . 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 480	. . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
31	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐶 ⊆ ∪ (𝐹‘𝑘))
32		boxcutc 8881	. . . . . . . . 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 585	. . . . . . . 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 8851	. . . . . . . . . 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 6833	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
36	35	unieqd 4875	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑥))
37		csbeq1a 3862	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → 𝐶 = ⦋𝑥 / 𝑘⦌𝐶)
38	36, 37	difeq12d 4078	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → (∪ (𝐹‘𝑘) ∖ 𝐶) = (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶))
39	38	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝑘 = 𝑥) → (∪ (𝐹‘𝑘) ∖ 𝐶) = (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶))
40	39	ifeq1da 4510	. . . . . . . . . 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 3056	. . . . . . . . 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 2774	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉)
45	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶Top)
46	1	ralrimiva 3127	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(𝐹‘𝑘)))
47		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝐶 ∈ (Clsd‘(𝐹‘𝑘))
48		nfcsb1v 3872	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶
49	48	nfel1 2914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥))
50		2fveq3 6838	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑥 → (Clsd‘(𝐹‘𝑘)) = (Clsd‘(𝐹‘𝑥)))
51	37, 50	eleq12d 2829	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → (𝐶 ∈ (Clsd‘(𝐹‘𝑘)) ↔ ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥))))
52	47, 49, 51	cbvralw 3277	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(𝐹‘𝑘)) ↔ ∀𝑥 ∈ 𝐴 ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)))
53	46, 52	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)))
54	53	r19.21bi 3227	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)))
55		eqid 2735	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑥) = ∪ (𝐹‘𝑥)
56	55	cldopn 22977	. . . . . . . . 9 ⊢ (⦋𝑥 / 𝑘⦌𝐶 ∈ (Clsd‘(𝐹‘𝑥)) → (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶) ∈ (𝐹‘𝑥))
57	54, 56	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶) ∈ (𝐹‘𝑥))
58	44, 45, 57	ptopn2 23530	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, (∪ (𝐹‘𝑥) ∖ ⦋𝑥 / 𝑘⦌𝐶), ∪ (𝐹‘𝑘)) ∈ (∏t‘𝐹))
59	43, 58	eqeltrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ (∏t‘𝐹) ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (∏t‘𝐹))
60		eqid 2735	. . . . . . . . 9 ⊢ ∪ (∏t‘𝐹) = ∪ (∏t‘𝐹)
61	60	iscld 22973	. . . . . . . 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 17	. . . . . . 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 480	. . . . . 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 714	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈ (Clsd‘(∏t‘𝐹)))
65	64	ralrimiva 3127	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘)) ∈ (Clsd‘(∏t‘𝐹)))
66	60	riincld 22990	. . . 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 585	. . 3 ⊢ (𝜑 → (∪ (∏t‘𝐹) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (Clsd‘(∏t‘𝐹)))
68	13, 67	eqeltrd 2835	. 2 ⊢ (𝜑 → (X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∩ ∩ 𝑥 ∈ 𝐴 X𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝐶, ∪ (𝐹‘𝑘))) ∈ (Clsd‘(∏t‘𝐹)))
69	7, 68	eqeltrd 2835	1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ⦋csb 3848   ∖ cdif 3897   ∩ cin 3899   ⊆ wss 3900  ifcif 4478  ∪ cuni 4862  ∩ ciin 4946  ⟶wf 6487  ‘cfv 6491  Xcixp 8837  ∏_tcpt 17360  Topctop 22839  Clsdccld 22962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-om 7809  df-1o 8397  df-2o 8398  df-ixp 8838  df-en 8886  df-fin 8889  df-fi 9316  df-topgen 17365  df-pt 17366  df-top 22840  df-bases 22892  df-cld 22965

This theorem is referenced by:  ptcldmpt  23560