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Mirrors > Home > MPE Home > Th. List > istps | Structured version Visualization version 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 21145 | . . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | |
2 | 1 | eleq2i 2851 | . 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}) |
3 | topontop 21125 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
4 | 0ntop 21117 | . . . . . 6 ⊢ ¬ ∅ ∈ Top | |
5 | istps.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐾) | |
6 | fvprc 6439 | . . . . . . . 8 ⊢ (¬ 𝐾 ∈ V → (TopOpen‘𝐾) = ∅) | |
7 | 5, 6 | syl5eq 2826 | . . . . . . 7 ⊢ (¬ 𝐾 ∈ V → 𝐽 = ∅) |
8 | 7 | eleq1d 2844 | . . . . . 6 ⊢ (¬ 𝐾 ∈ V → (𝐽 ∈ Top ↔ ∅ ∈ Top)) |
9 | 4, 8 | mtbiri 319 | . . . . 5 ⊢ (¬ 𝐾 ∈ V → ¬ 𝐽 ∈ Top) |
10 | 9 | con4i 114 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ V) |
11 | 3, 10 | syl 17 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V) |
12 | fveq2 6446 | . . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
13 | 12, 5 | syl6eqr 2832 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
14 | fveq2 6446 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
15 | istps.a | . . . . . 6 ⊢ 𝐴 = (Base‘𝐾) | |
16 | 14, 15 | syl6eqr 2832 | . . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴) |
17 | 16 | fveq2d 6450 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴)) |
18 | 13, 17 | eleq12d 2853 | . . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴))) |
19 | 11, 18 | elab3 3566 | . 2 ⊢ (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴)) |
20 | 2, 19 | bitri 267 | 1 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 = wceq 1601 ∈ wcel 2107 {cab 2763 Vcvv 3398 ∅c0 4141 ‘cfv 6135 Basecbs 16255 TopOpenctopn 16468 Topctop 21105 TopOnctopon 21122 TopSpctps 21144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-top 21106 df-topon 21123 df-topsp 21145 |
This theorem is referenced by: istps2 21147 tpspropd 21150 tsettps 21153 indistps2ALT 21226 resstps 21399 prdstps 21841 imastps 21933 xpstopnlem2 22023 tmdtopon 22293 tgptopon 22294 istgp2 22303 oppgtmd 22309 distgp 22311 indistgp 22312 symgtgp 22313 qustgplem 22332 prdstmdd 22335 eltsms 22344 tsmscls 22349 tsmsgsum 22350 tsmsid 22351 tsmsmhm 22357 tsmsadd 22358 dvrcn 22395 cnmpt1vsca 22405 cnmpt2vsca 22406 tlmtgp 22407 ressusp 22477 tustps 22485 ucncn 22497 neipcfilu 22508 cnextucn 22515 ucnextcn 22516 isxms2 22661 ressxms 22738 prdsxmslem2 22742 nrgtrg 22902 cnfldtopon 22994 cnmpt1ds 23053 cnmpt2ds 23054 nmcn 23055 cnmpt1ip 23453 cnmpt2ip 23454 csscld 23455 clsocv 23456 minveclem4a 23636 mhmhmeotmd 30571 rrxtopon 41436 qndenserrnopnlem 41445 |
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