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Theorem topnid 17357

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 6847	. . 3 ⊢ 𝐵 ∈ V
3		restid2 17352	. . 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 17356	. 2 ⊢ (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊)
7	4, 6	eqtr3di 2785	1 ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3439 ⊆ wss 3900 𝒫 cpw 4553 ‘cfv 6491 (class class class)co 7358 Basecbs 17138 TopSetcts 17185 ↾_t crest 17342 TopOpenctopn 17343

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-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-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5518 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-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-rest 17344 df-topn 17345

This theorem is referenced by: topontopn 22886 prdstopn 23574 imastopn 23666 setsmstopn 24424 tngtopn 24596 circtopn 33973 rspectopn 34003

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