Theorem cncls 23431

Description: Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
cncls	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cncls

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnf2 23406	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expia 1139	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌))
3		elpwi 4569	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
4	3	adantl 486	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ 𝑋)
5		toponuni 23071	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	5	ad2antrr 738	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 = ∪ 𝐽)
7	4, 6	sseqtrd 3973	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ ∪ 𝐽)
8		eqid 2763	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	cnclsi 23429	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ⊆ ∪ 𝐽) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))
10	9	expcom 418	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐽 → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))))
11	7, 10	syl 18	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))))
12	11	ralrimdva 3165	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))))
13	2, 12	jcad 521	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))))
14		toponmax 23083	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
15	14	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 ∈ 𝐽)
16		cnvimass 6084	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
17		fdm 6715	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
18	17	ad2antlr 739	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝑋)
19	16, 18	sseqtrid 3979	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ⊆ 𝑋)
20	15, 19	sselpwd 5299	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ∈ 𝒫 𝑋)
21		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((cls‘𝐽)‘𝑥) = ((cls‘𝐽)‘(◡𝐹 “ 𝑦)))
22	21	imaeq2d 6062	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ ((cls‘𝐽)‘𝑥)) = (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))))
23		imaeq2 6058	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)))
24	23	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((cls‘𝐾)‘(𝐹 “ 𝑥)) = ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))))
25	22, 24	sseq12d 3970	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) ↔ (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦)))))
26	25	rspcv 3577	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) → (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦)))))
27	20, 26	syl 18	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) → (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦)))))
28		topontop 23070	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
29	28	ad3antlr 743	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐾 ∈ Top)
30		elpwi 4569	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑌 → 𝑦 ⊆ 𝑌)
31	30	adantl 486	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 ⊆ 𝑌)
32		toponuni 23071	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
33	32	ad3antlr 743	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑌 = ∪ 𝐾)
34	31, 33	sseqtrd 3973	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 ⊆ ∪ 𝐾)
35		ffun 6708	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
36	35	ad2antlr 739	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → Fun 𝐹)
37		funimacnv 6617	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝑦)) = (𝑦 ∩ ran 𝐹))
38	36, 37	syl 18	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = (𝑦 ∩ ran 𝐹))
39		inss1 4189	. . . . . . . . . 10 ⊢ (𝑦 ∩ ran 𝐹) ⊆ 𝑦
40	38, 39	eqsstrdi 3981	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) ⊆ 𝑦)
41		eqid 2763	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
42	41	clsss 23211	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾 ∧ (𝐹 “ (◡𝐹 “ 𝑦)) ⊆ 𝑦) → ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
43	29, 34, 40, 42	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
44		sstr2 3944	. . . . . . . 8 ⊢ ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) → (((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦) → (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦)))
45	43, 44	syl5com 32	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) → (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦)))
46		topontop 23070	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
47	46	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐽 ∈ Top)
48	5	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 = ∪ 𝐽)
49	18, 48	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = ∪ 𝐽)
50	16, 49	sseqtrid 3979	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)
51	8	clsss3 23216	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ ∪ 𝐽)
52	47, 50, 51	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ ∪ 𝐽)
53	52, 49	sseqtrrd 3974	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ dom 𝐹)
54		funimass3 7049	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦))))
55	36, 53, 54	syl2anc 595	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦) ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦))))
56	45, 55	sylibd 242	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦))))
57	27, 56	syld 48	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦))))
58	57	ralrimdva 3165	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)) → ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦))))
59	58	imdistanda 581	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦)))))
60		cncls2 23430	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑦)))))
61	59, 60	sylibrd 262	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾)))
62	13, 61	impbid 215	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   “ cima 5664  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Topctop 23050  TopOnctopon 23067  clsccl 23175   Cn ccn 23381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8822  df-top 23051  df-topon 23068  df-cld 23176  df-cls 23178  df-cn 23384

This theorem is referenced by: (None)