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Theorem topnid 17487

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 6895	. . 3 ⊢ 𝐵 ∈ V
3		restid2 17482	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐽 ⊆ 𝒫 𝐵) → (𝐽 ↾_t 𝐵) = 𝐽)
4	2, 3	mpan 702	. 2 ⊢ (𝐽 ⊆ 𝒫 𝐵 → (𝐽 ↾_t 𝐵) = 𝐽)
5		topnval.2	. . 3 ⊢ 𝐽 = (TopSet‘𝑊)
6	1, 5	topnval 17486	. 2 ⊢ (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊)
7	4, 6	eqtr3di 2811	1 ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1568 ∈ wcel 2141 Vcvv 3453 ⊆ wss 3904 𝒫 cpw 4561 ‘cfv 6536 (class class class)co 7410 Basecbs 17268 TopSetcts 17315 ↾_t crest 17472 TopOpenctopn 17473

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 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 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 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-1st 7985 df-2nd 7986 df-rest 17474 df-topn 17475

This theorem is referenced by: topontopn 23076 prdstopn 23764 imastopn 23856 setsmstopn 24614 tngtopn 24786 circtopn 34193 rspectopn 34223

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