Theorem topnval 17423

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 6902	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2		topnval.2	. . . . . 6 ⊢ 𝐽 = (TopSet‘𝑊)
3	1, 2	eqtr4di 2786	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4		fveq2 6902	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		topnval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
6	4, 5	eqtr4di 2786	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
7	3, 6	oveq12d 7444	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵))
8		df-topn 17412	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
9		ovex 7459	. . . 4 ⊢ (𝐽 ↾t 𝐵) ∈ V
10	7, 8, 9	fvmpt 7010	. . 3 ⊢ (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵))
11	10	eqcomd 2734	. 2 ⊢ (𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))
12		0rest 17418	. . 3 ⊢ (∅ ↾t 𝐵) = ∅
13		fvprc 6894	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
14	2, 13	eqtrid 2780	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐽 = ∅)
15	14	oveq1d 7441	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (∅ ↾t 𝐵))
16		fvprc 6894	. . 3 ⊢ (¬ 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
17	12, 15, 16	3eqtr4a 2794	. 2 ⊢ (¬ 𝑊 ∈ V → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))
18	11, 17	pm2.61i 182	1 ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3473  ∅c0 4326  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  TopSetcts 17246   ↾_t crest 17409  TopOpenctopn 17410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-rest 17411  df-topn 17412

This theorem is referenced by:  topnid  17424  topnpropd  17425  efmndtopn  18842  oppgtopn  19314  symgtopn  19368  mgptopn  20093  resstopn  23110  prdstopn  23552  tuslem  24191  tuslemOLD  24192  xrge0tsms  24770  om1opn  24983  xrge0tsmsd  32792  xrge0tmdALT  33580