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Theorem topnval 17338

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 6822	. . . . . 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 6822	. . . . . 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 7367	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾_t (Base‘𝑤)) = (𝐽 ↾_t 𝐵))
8		df-topn 17327	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾_t (Base‘𝑤)))
9		ovex 7382	. . . 4 ⊢ (𝐽 ↾_t 𝐵) ∈ V
10	7, 8, 9	fvmpt 6930	. . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾_t 𝐵))
11	10	eqcomd 2735	. 2 ⊢ (𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊))
12		0rest 17333	. . 3 ⊢ (∅ ↾_t 𝐵) = ∅
13		fvprc 6814	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
14	2, 13	eqtrid 2776	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅)
15	14	oveq1d 7364	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (∅ ↾_t 𝐵))
16		fvprc 6814	. . 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 1540 ∈ wcel 2109 Vcvv 3436 ∅c0 4284 ‘cfv 6482 (class class class)co 7349 Basecbs 17120 TopSetcts 17167 ↾_t crest 17324 TopOpenctopn 17325

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-rest 17326 df-topn 17327

This theorem is referenced by: topnid 17339 topnpropd 17340 efmndtopn 18757 oppgtopn 19232 symgtopn 19285 mgptopn 20033 resstopn 23071 prdstopn 23513 tuslem 24152 xrge0tsms 24721 om1opn 24934 xrge0tsmsd 33015 xrge0tmdALT 33913

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