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Mirrors > Home > ILE Home > Th. List > istps | GIF version |
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
istps.a | ⊢ 𝐴 = (Base‘𝐾) |
istps.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
Ref | Expression |
---|---|
istps | ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-topsp 12576 | . . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | |
2 | 1 | eleq2i 2231 | . 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}) |
3 | topontop 12559 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
4 | topnfn 12503 | . . . . . . 7 ⊢ TopOpen Fn V | |
5 | fnrel 5280 | . . . . . . 7 ⊢ (TopOpen Fn V → Rel TopOpen) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ Rel TopOpen |
7 | 0opn 12551 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
8 | istps.j | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐾) | |
9 | 7, 8 | eleqtrdi 2257 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopOpen‘𝐾)) |
10 | relelfvdm 5512 | . . . . . 6 ⊢ ((Rel TopOpen ∧ ∅ ∈ (TopOpen‘𝐾)) → 𝐾 ∈ dom TopOpen) | |
11 | 6, 9, 10 | sylancr 411 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopOpen) |
12 | 11 | elexd 2734 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ V) |
13 | 3, 12 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V) |
14 | fveq2 5480 | . . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
15 | 14, 8 | eqtr4di 2215 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
16 | fveq2 5480 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
17 | istps.a | . . . . . 6 ⊢ 𝐴 = (Base‘𝐾) | |
18 | 16, 17 | eqtr4di 2215 | . . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴) |
19 | 18 | fveq2d 5484 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴)) |
20 | 15, 19 | eleq12d 2235 | . . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴))) |
21 | 13, 20 | elab3 2873 | . 2 ⊢ (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴)) |
22 | 2, 21 | bitri 183 | 1 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 ∈ wcel 2135 {cab 2150 Vcvv 2721 ∅c0 3404 dom cdm 4598 Rel wrel 4603 Fn wfn 5177 ‘cfv 5182 Basecbs 12337 TopOpenctopn 12499 Topctop 12542 TopOnctopon 12555 TopSpctps 12575 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-ndx 12340 df-slot 12341 df-base 12343 df-tset 12418 df-rest 12500 df-topn 12501 df-top 12543 df-topon 12556 df-topsp 12576 |
This theorem is referenced by: istps2 12578 tpspropd 12581 tsettps 12583 isxms2 12999 |
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