Theorem cncls 23252

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 23227	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expia 1122	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌))
3		elpwi 4549	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
4	3	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ 𝑋)
5		toponuni 22892	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	5	ad2antrr 727	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 = ∪ 𝐽)
7	4, 6	sseqtrd 3959	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ ∪ 𝐽)
8		eqid 2737	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	cnclsi 23250	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ⊆ ∪ 𝐽) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))
10	9	expcom 413	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐽 → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))))
11	7, 10	syl 17	. . . 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 3138	. . 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 512	. 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 22904	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
15	14	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 ∈ 𝐽)
16		cnvimass 6042	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
17		fdm 6672	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
18	17	ad2antlr 728	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = 𝑋)
19	16, 18	sseqtrid 3965	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ⊆ 𝑋)
20	15, 19	sselpwd 5266	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ∈ 𝒫 𝑋)
21		fveq2 6835	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((cls‘𝐽)‘𝑥) = ((cls‘𝐽)‘(◡𝐹 “ 𝑦)))
22	21	imaeq2d 6020	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ ((cls‘𝐽)‘𝑥)) = (𝐹 “ ((cls‘𝐽)‘(◡𝐹 “ 𝑦))))
23		imaeq2 6016	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)))
24	23	fveq2d 6839	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((cls‘𝐾)‘(𝐹 “ 𝑥)) = ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))))
25	22, 24	sseq12d 3956	. . . . . . . 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 3561	. . . . . . 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 17	. . . . . 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 22891	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
29	28	ad3antlr 732	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐾 ∈ Top)
30		elpwi 4549	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑌 → 𝑦 ⊆ 𝑌)
31	30	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 ⊆ 𝑌)
32		toponuni 22892	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
33	32	ad3antlr 732	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑌 = ∪ 𝐾)
34	31, 33	sseqtrd 3959	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑦 ⊆ ∪ 𝐾)
35		ffun 6666	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
36	35	ad2antlr 728	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → Fun 𝐹)
37		funimacnv 6574	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝑦)) = (𝑦 ∩ ran 𝐹))
38	36, 37	syl 17	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = (𝑦 ∩ ran 𝐹))
39		inss1 4178	. . . . . . . . . 10 ⊢ (𝑦 ∩ ran 𝐹) ⊆ 𝑦
40	38, 39	eqsstrdi 3967	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) ⊆ 𝑦)
41		eqid 2737	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
42	41	clsss 23032	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾 ∧ (𝐹 “ (◡𝐹 “ 𝑦)) ⊆ 𝑦) → ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
43	29, 34, 40, 42	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐾)‘(𝐹 “ (◡𝐹 “ 𝑦))) ⊆ ((cls‘𝐾)‘𝑦))
44		sstr2 3929	. . . . . . . 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 31	. . . . . . 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 22891	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
47	46	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝐽 ∈ Top)
48	5	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → 𝑋 = ∪ 𝐽)
49	18, 48	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → dom 𝐹 = ∪ 𝐽)
50	16, 49	sseqtrid 3965	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)
51	8	clsss3 23037	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ ∪ 𝐽)
52	47, 50, 51	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ ∪ 𝐽)
53	52, 49	sseqtrrd 3960	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ 𝒫 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑦)) ⊆ dom 𝐹)
54		funimass3 7001	. . . . . . . 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 585	. . . . . . 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 239	. . . . . 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 47	. . . . 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 3138	. . . 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 571	. . 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 23251	. . 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 259	. 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 212	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   “ cima 5628  Fun wfun 6487  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  Topctop 22871  TopOnctopon 22888  clsccl 22996   Cn ccn 23202

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 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8769  df-top 22872  df-topon 22889  df-cld 22997  df-cls 22999  df-cn 23205

This theorem is referenced by: (None)