Theorem topnvalg 11831

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 2644	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		restfn 11823	. . . 4 ⊢ ↾t Fn (V × V)
3		topnval.2	. . . . 5 ⊢ 𝐽 = (TopSet‘𝑊)
4		tsetslid 11808	. . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
5	4	slotex 11685	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (TopSet‘𝑊) ∈ V)
6	3, 5	syl5eqel 2181	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐽 ∈ V)
7		topnval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
8		baseslid 11714	. . . . . 6 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
9	8	slotex 11685	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (Base‘𝑊) ∈ V)
10	7, 9	syl5eqel 2181	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐵 ∈ V)
11		fnovex 5720	. . . 4 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽 ↾t 𝐵) ∈ V)
12	2, 6, 10, 11	mp3an2i 1285	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) ∈ V)
13		fveq2 5340	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
14	13, 3	syl6eqr 2145	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
15		fveq2 5340	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16	15, 7	syl6eqr 2145	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
17	14, 16	oveq12d 5708	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵))
18		df-topn 11822	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
19	17, 18	fvmptg 5415	. . 3 ⊢ ((𝑊 ∈ V ∧ (𝐽 ↾t 𝐵) ∈ V) → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵))
20	1, 12, 19	syl2anc 404	. 2 ⊢ (𝑊 ∈ 𝑉 → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵))
21	20	eqcomd 2100	1 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))

Syntax hints:   → wi 4   = wceq 1296   ∈ wcel 1445  Vcvv 2633   × cxp 4465   Fn wfn 5044  ‘cfv 5049  (class class class)co 5690  Basecbs 11658  TopSetcts 11726   ↾_t crest 11819  TopOpenctopn 11820

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-cnex 7533  ax-resscn 7534  ax-1re 7536  ax-addrcl 7539

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-5 8582  df-6 8583  df-7 8584  df-8 8585  df-9 8586  df-ndx 11661  df-slot 11662  df-base 11664  df-tset 11739  df-rest 11821  df-topn 11822

This theorem is referenced by:  topnidg  11832  topnpropgd  11833