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Mirrors > Home > ILE Home > Th. List > topnvalg | GIF version |
Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.) |
Ref | Expression |
---|---|
topnval.1 | ⊢ 𝐵 = (Base‘𝑊) |
topnval.2 | ⊢ 𝐽 = (TopSet‘𝑊) |
Ref | Expression |
---|---|
topnvalg | ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
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)) |
Colors of variables: wff set class |
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 |
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