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Theorem topnid 17417

Description: Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)

**Hypotheses**
Ref	Expression
topnval.1	⊢ 𝐵 = (Base‘𝑊)
topnval.2	⊢ 𝐽 = (TopSet‘𝑊)

**Assertion**
Ref	Expression
topnid	⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊))

**Proof of Theorem topnid**
Step	Hyp	Ref	Expression
1		topnval.1	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
2	1	fvexi 6911	. . 3 ⊢ 𝐵 ∈ V
3		restid2 17412	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐽 ⊆ 𝒫 𝐵) → (𝐽 ↾_t 𝐵) = 𝐽)
4	2, 3	mpan 689	. 2 ⊢ (𝐽 ⊆ 𝒫 𝐵 → (𝐽 ↾_t 𝐵) = 𝐽)
5		topnval.2	. . 3 ⊢ 𝐽 = (TopSet‘𝑊)
6	1, 5	topnval 17416	. 2 ⊢ (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊)
7	4, 6	eqtr3di 2783	1 ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 𝒫 cpw 4603 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 TopSetcts 17239 ↾_t crest 17402 TopOpenctopn 17403

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-rest 17404 df-topn 17405

This theorem is referenced by: topontopn 22855 prdstopn 23545 imastopn 23637 setsmstopn 24399 tngtopn 24580 circtopn 33438 rspectopn 33468

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