topnid - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > topnid		Structured version Visualization version GIF version

Theorem topnid 17482

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 6921	. . 3 ⊢ 𝐵 ∈ V
3		restid2 17477	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐽 ⊆ 𝒫 𝐵) → (𝐽 ↾_t 𝐵) = 𝐽)
4	2, 3	mpan 690	. 2 ⊢ (𝐽 ⊆ 𝒫 𝐵 → (𝐽 ↾_t 𝐵) = 𝐽)
5		topnval.2	. . 3 ⊢ 𝐽 = (TopSet‘𝑊)
6	1, 5	topnval 17481	. 2 ⊢ (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊)
7	4, 6	eqtr3di 2790	1 ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 TopSetcts 17304 ↾_t crest 17467 TopOpenctopn 17468

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-rest 17469 df-topn 17470

This theorem is referenced by: topontopn 22962 prdstopn 23652 imastopn 23744 setsmstopn 24506 tngtopn 24687 circtopn 33798 rspectopn 33828

W3C validator