![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2763 | . . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | restfn 12748 | . . . 4 ⊢ ↾t Fn (V × V) | |
3 | topnval.2 | . . . . 5 ⊢ 𝐽 = (TopSet‘𝑊) | |
4 | tsetslid 12699 | . . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
5 | 4 | slotex 12539 | . . . . 5 ⊢ (𝑊 ∈ 𝑉 → (TopSet‘𝑊) ∈ V) |
6 | 3, 5 | eqeltrid 2276 | . . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐽 ∈ V) |
7 | topnval.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
8 | baseslid 12569 | . . . . . 6 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | |
9 | 8 | slotex 12539 | . . . . 5 ⊢ (𝑊 ∈ 𝑉 → (Base‘𝑊) ∈ V) |
10 | 7, 9 | eqeltrid 2276 | . . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐵 ∈ V) |
11 | fnovex 5929 | . . . 4 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽 ↾t 𝐵) ∈ V) | |
12 | 2, 6, 10, 11 | mp3an2i 1353 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) ∈ V) |
13 | fveq2 5534 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊)) | |
14 | 13, 3 | eqtr4di 2240 | . . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽) |
15 | fveq2 5534 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
16 | 15, 7 | eqtr4di 2240 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
17 | 14, 16 | oveq12d 5914 | . . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵)) |
18 | df-topn 12747 | . . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | |
19 | 17, 18 | fvmptg 5613 | . . 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 2195 | 1 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 × cxp 4642 Fn wfn 5230 ‘cfv 5235 (class class class)co 5896 Basecbs 12512 TopSetcts 12595 ↾t crest 12744 TopOpenctopn 12745 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-cnex 7932 ax-resscn 7933 ax-1re 7935 ax-addrcl 7938 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-7 9013 df-8 9014 df-9 9015 df-ndx 12515 df-slot 12516 df-base 12518 df-tset 12608 df-rest 12746 df-topn 12747 |
This theorem is referenced by: topnidg 12757 topnpropgd 12758 mgptopng 13283 |
Copyright terms: Public domain | W3C validator |