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Theorem topnval 16095

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 6191	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2		topnval.2	. . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊)
3	1, 2	syl6eqr 2674	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		topnval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
6	4, 5	syl6eqr 2674	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
7	3, 6	oveq12d 6668	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾_t (Base‘𝑤)) = (𝐽 ↾_t 𝐵))
8		df-topn 16084	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾_t (Base‘𝑤)))
9		ovex 6678	. . . 4 ⊢ (𝐽 ↾_t 𝐵) ∈ V
10	7, 8, 9	fvmpt 6282	. . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾_t 𝐵))
11	10	eqcomd 2628	. 2 ⊢ (𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊))
12		0rest 16090	. . 3 ⊢ (∅ ↾_t 𝐵) = ∅
13		fvprc 6185	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
14	2, 13	syl5eq 2668	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅)
15	14	oveq1d 6665	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (∅ ↾_t 𝐵))
16		fvprc 6185	. . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
17	12, 15, 16	3eqtr4a 2682	. 2 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊))
18	11, 17	pm2.61i 176	1 ⊢ (𝐽 ↾_t 𝐵) = (TopOpen‘𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 TopSetcts 15947 ↾_t crest 16081 TopOpenctopn 16082

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-rest 16083 df-topn 16084

This theorem is referenced by: topnid 16096 topnpropd 16097 oppgtopn 17783 symgtopn 17825 mgptopn 18498 resstopn 20990 prdstopn 21431 tuslem 22071 xrge0tsms 22637 om1opn 22836 xrge0tsmsd 29785 xrge0tmdOLD 29991

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