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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 12187 | . . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | |
2 | 1 | eleq2i 2204 | . 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}) |
3 | topontop 12170 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
4 | topnfn 12114 | . . . . . . 7 ⊢ TopOpen Fn V | |
5 | fnrel 5216 | . . . . . . 7 ⊢ (TopOpen Fn V → Rel TopOpen) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ Rel TopOpen |
7 | 0opn 12162 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
8 | istps.j | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐾) | |
9 | 7, 8 | eleqtrdi 2230 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopOpen‘𝐾)) |
10 | relelfvdm 5446 | . . . . . 6 ⊢ ((Rel TopOpen ∧ ∅ ∈ (TopOpen‘𝐾)) → 𝐾 ∈ dom TopOpen) | |
11 | 6, 9, 10 | sylancr 410 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopOpen) |
12 | 11 | elexd 2694 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ V) |
13 | 3, 12 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V) |
14 | fveq2 5414 | . . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
15 | 14, 8 | syl6eqr 2188 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
16 | fveq2 5414 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
17 | istps.a | . . . . . 6 ⊢ 𝐴 = (Base‘𝐾) | |
18 | 16, 17 | syl6eqr 2188 | . . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴) |
19 | 18 | fveq2d 5418 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴)) |
20 | 15, 19 | eleq12d 2208 | . . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴))) |
21 | 13, 20 | elab3 2831 | . 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 1331 ∈ wcel 1480 {cab 2123 Vcvv 2681 ∅c0 3358 dom cdm 4534 Rel wrel 4539 Fn wfn 5113 ‘cfv 5118 Basecbs 11948 TopOpenctopn 12110 Topctop 12153 TopOnctopon 12166 TopSpctps 12186 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-7 8777 df-8 8778 df-9 8779 df-ndx 11951 df-slot 11952 df-base 11954 df-tset 12029 df-rest 12111 df-topn 12112 df-top 12154 df-topon 12167 df-topsp 12187 |
This theorem is referenced by: istps2 12189 tpspropd 12192 tsettps 12194 isxms2 12610 |
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