Theorem topnvalg 12577

Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)

Hypotheses

Ref	Expression
topnval.1	⊢ 𝐵 = (Base‘𝑊)
topnval.2	⊢ 𝐽 = (TopSet‘𝑊)

Assertion

Ref	Expression
topnvalg	⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))

Proof of Theorem topnvalg

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2741	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		restfn 12569	. . . 4 ⊢ ↾t Fn (V × V)
3		topnval.2	. . . . 5 ⊢ 𝐽 = (TopSet‘𝑊)
4		tsetslid 12554	. . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
5	4	slotex 12430	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (TopSet‘𝑊) ∈ V)
6	3, 5	eqeltrid 2257	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐽 ∈ V)
7		topnval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
8		baseslid 12459	. . . . . 6 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
9	8	slotex 12430	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (Base‘𝑊) ∈ V)
10	7, 9	eqeltrid 2257	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐵 ∈ V)
11		fnovex 5883	. . . 4 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽 ↾t 𝐵) ∈ V)
12	2, 6, 10, 11	mp3an2i 1337	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) ∈ V)
13		fveq2 5494	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
14	13, 3	eqtr4di 2221	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
15		fveq2 5494	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16	15, 7	eqtr4di 2221	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
17	14, 16	oveq12d 5868	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵))
18		df-topn 12568	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
19	17, 18	fvmptg 5570	. . 3 ⊢ ((𝑊 ∈ V ∧ (𝐽 ↾t 𝐵) ∈ V) → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵))
20	1, 12, 19	syl2anc 409	. 2 ⊢ (𝑊 ∈ 𝑉 → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵))
21	20	eqcomd 2176	1 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  Vcvv 2730   × cxp 4607   Fn wfn 5191  ‘cfv 5196  (class class class)co 5850  Basecbs 12403  TopSetcts 12472   ↾_t crest 12565  TopOpenctopn 12566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-cnex 7852  ax-resscn 7853  ax-1re 7855  ax-addrcl 7858

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-5 8927  df-6 8928  df-7 8929  df-8 8930  df-9 8931  df-ndx 12406  df-slot 12407  df-base 12409  df-tset 12485  df-rest 12567  df-topn 12568

This theorem is referenced by:  topnidg  12578  topnpropgd  12579