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Theorem topnval 17379

Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)

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

**Assertion**
Ref	Expression
topnval	⊢ (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊)

Proof of Theorem topnval Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2		topnval.2	. . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊)
3	1, 2	eqtr4di 2782	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4		fveq2 6881	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		topnval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
6	4, 5	eqtr4di 2782	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
7	3, 6	oveq12d 7419	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾_t (Base‘𝑤)) = (𝐽 ↾_t 𝐵))
8		df-topn 17368	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾_t (Base‘𝑤)))
9		ovex 7434	. . . 4 ⊢ (𝐽 ↾_t 𝐵) ∈ V
10	7, 8, 9	fvmpt 6988	. . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾_t 𝐵))
11	10	eqcomd 2730	. 2 ⊢ (𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊))
12		0rest 17374	. . 3 ⊢ (∅ ↾_t 𝐵) = ∅
13		fvprc 6873	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
14	2, 13	eqtrid 2776	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅)
15	14	oveq1d 7416	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (∅ ↾_t 𝐵))
16		fvprc 6873	. . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
17	12, 15, 16	3eqtr4a 2790	. 2 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊))
18	11, 17	pm2.61i 182	1 ⊢ (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∅c0 4314 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 TopSetcts 17202 ↾_t crest 17365 TopOpenctopn 17366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-rest 17367 df-topn 17368

This theorem is referenced by: topnid 17380 topnpropd 17381 efmndtopn 18798 oppgtopn 19262 symgtopn 19316 mgptopn 20041 resstopn 23012 prdstopn 23454 tuslem 24093 tuslemOLD 24094 xrge0tsms 24672 om1opn 24885 xrge0tsmsd 32677 xrge0tmdALT 33415

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