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Theorem topnid 17356

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 6846	. . 3 ⊢ 𝐵 ∈ V
3		restid2 17351	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐽 ⊆ 𝒫 𝐵) → (𝐽 ↾_t 𝐵) = 𝐽)
4	2, 3	mpan 691	. 2 ⊢ (𝐽 ⊆ 𝒫 𝐵 → (𝐽 ↾_t 𝐵) = 𝐽)
5		topnval.2	. . 3 ⊢ 𝐽 = (TopSet‘𝑊)
6	1, 5	topnval 17355	. 2 ⊢ (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊)
7	4, 6	eqtr3di 2787	1 ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ⊆ wss 3890 𝒫 cpw 4542 ‘cfv 6490 (class class class)co 7358 Basecbs 17137 TopSetcts 17184 ↾_t crest 17341 TopOpenctopn 17342

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680

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-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-rest 17343 df-topn 17344

This theorem is referenced by: topontopn 22883 prdstopn 23571 imastopn 23663 setsmstopn 24421 tngtopn 24593 circtopn 33987 rspectopn 34017

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