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Theorem topnval 17479

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 6906	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2		topnval.2	. . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊)
3	1, 2	eqtr4di 2795	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4		fveq2 6906	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		topnval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
6	4, 5	eqtr4di 2795	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
7	3, 6	oveq12d 7449	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾_t (Base‘𝑤)) = (𝐽 ↾_t 𝐵))
8		df-topn 17468	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾_t (Base‘𝑤)))
9		ovex 7464	. . . 4 ⊢ (𝐽 ↾_t 𝐵) ∈ V
10	7, 8, 9	fvmpt 7016	. . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾_t 𝐵))
11	10	eqcomd 2743	. 2 ⊢ (𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊))
12		0rest 17474	. . 3 ⊢ (∅ ↾_t 𝐵) = ∅
13		fvprc 6898	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
14	2, 13	eqtrid 2789	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅)
15	14	oveq1d 7446	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (∅ ↾_t 𝐵))
16		fvprc 6898	. . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
17	12, 15, 16	3eqtr4a 2803	. 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 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TopSetcts 17303 ↾_t crest 17465 TopOpenctopn 17466

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-rest 17467 df-topn 17468

This theorem is referenced by: topnid 17480 topnpropd 17481 efmndtopn 18896 oppgtopn 19372 symgtopn 19424 mgptopn 20145 resstopn 23194 prdstopn 23636 tuslem 24275 tuslemOLD 24276 xrge0tsms 24856 om1opn 25069 xrge0tsmsd 33065 xrge0tmdALT 33945

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