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Theorem topnval 17448

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 6876	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2		topnval.2	. . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊)
3	1, 2	eqtr4di 2788	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4		fveq2 6876	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		topnval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
6	4, 5	eqtr4di 2788	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
7	3, 6	oveq12d 7423	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾_t (Base‘𝑤)) = (𝐽 ↾_t 𝐵))
8		df-topn 17437	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾_t (Base‘𝑤)))
9		ovex 7438	. . . 4 ⊢ (𝐽 ↾_t 𝐵) ∈ V
10	7, 8, 9	fvmpt 6986	. . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾_t 𝐵))
11	10	eqcomd 2741	. 2 ⊢ (𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊))
12		0rest 17443	. . 3 ⊢ (∅ ↾_t 𝐵) = ∅
13		fvprc 6868	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
14	2, 13	eqtrid 2782	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅)
15	14	oveq1d 7420	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (∅ ↾_t 𝐵))
16		fvprc 6868	. . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
17	12, 15, 16	3eqtr4a 2796	. 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 1540 ∈ wcel 2108 Vcvv 3459 ∅c0 4308 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 TopSetcts 17277 ↾_t crest 17434 TopOpenctopn 17435

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-rest 17436 df-topn 17437

This theorem is referenced by: topnid 17449 topnpropd 17450 efmndtopn 18861 oppgtopn 19336 symgtopn 19387 mgptopn 20108 resstopn 23124 prdstopn 23566 tuslem 24205 xrge0tsms 24774 om1opn 24987 xrge0tsmsd 33056 xrge0tmdALT 33977

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