Theorem topnvalg 12705

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 2750	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		restfn 12697	. . . 4 ⊢ ↾t Fn (V × V)
3		topnval.2	. . . . 5 ⊢ 𝐽 = (TopSet‘𝑊)
4		tsetslid 12648	. . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
5	4	slotex 12491	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (TopSet‘𝑊) ∈ V)
6	3, 5	eqeltrid 2264	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐽 ∈ V)
7		topnval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
8		baseslid 12521	. . . . . 6 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
9	8	slotex 12491	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (Base‘𝑊) ∈ V)
10	7, 9	eqeltrid 2264	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐵 ∈ V)
11		fnovex 5910	. . . 4 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽 ↾t 𝐵) ∈ V)
12	2, 6, 10, 11	mp3an2i 1342	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) ∈ V)
13		fveq2 5517	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
14	13, 3	eqtr4di 2228	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
15		fveq2 5517	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16	15, 7	eqtr4di 2228	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
17	14, 16	oveq12d 5895	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵))
18		df-topn 12696	. . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
19	17, 18	fvmptg 5594	. . 3 ⊢ ((𝑊 ∈ V ∧ (𝐽 ↾t 𝐵) ∈ V) → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵))
20	1, 12, 19	syl2anc 411	. 2 ⊢ (𝑊 ∈ 𝑉 → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵))
21	20	eqcomd 2183	1 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2739   × cxp 4626   Fn wfn 5213  ‘cfv 5218  (class class class)co 5877  Basecbs 12464  TopSetcts 12544   ↾_t crest 12693  TopOpenctopn 12694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-7 8985  df-8 8986  df-9 8987  df-ndx 12467  df-slot 12468  df-base 12470  df-tset 12557  df-rest 12695  df-topn 12696

This theorem is referenced by:  topnidg  12706  topnpropgd  12707  mgptopng  13144