Theorem topnval 17463

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 6867	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2		topnval.2	. . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊)
3	1, 2	eqtr4di 2815	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4		fveq2 6867	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		topnval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
6	4, 5	eqtr4di 2815	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
7	3, 6	oveq12d 7414	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵))
8		df-topn 17452	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
9		ovex 7429	. . . 4 ⊢ (𝐽 ↾t 𝐵) ∈ V
10	7, 8, 9	fvmpt 6975	. . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵))
11	10	eqcomd 2768	. 2 ⊢ (𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))
12		0rest 17458	. . 3 ⊢ (∅ ↾t 𝐵) = ∅
13		fvprc 6859	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
14	2, 13	eqtrid 2809	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅)
15	14	oveq1d 7411	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (∅ ↾t 𝐵))
16		fvprc 6859	. . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
17	12, 15, 16	3eqtr4a 2823	. 2 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))
18	11, 17	pm2.61i 183	1 ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)

Syntax hints:  ¬ wn 3   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  TopSetcts 17292   ↾_t crest 17449  TopOpenctopn 17450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-rest 17451  df-topn 17452

This theorem is referenced by:  topnid  17464  topnpropd  17465  efmndtopn  18917  oppgtopn  19393  symgtopn  19446  mgptopn  20194  resstopn  23243  prdstopn  23685  tuslem  24323  xrge0tsms  24892  om1opn  25095  xrge0tsmsd  33250  xrge0tmdALT  34240