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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 21544 | . . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | |
2 | 1 | eleq2i 2907 | . 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}) |
3 | topontop 21524 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
4 | 0ntop 21516 | . . . . . 6 ⊢ ¬ ∅ ∈ Top | |
5 | istps.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐾) | |
6 | fvprc 6666 | . . . . . . . 8 ⊢ (¬ 𝐾 ∈ V → (TopOpen‘𝐾) = ∅) | |
7 | 5, 6 | syl5eq 2871 | . . . . . . 7 ⊢ (¬ 𝐾 ∈ V → 𝐽 = ∅) |
8 | 7 | eleq1d 2900 | . . . . . 6 ⊢ (¬ 𝐾 ∈ V → (𝐽 ∈ Top ↔ ∅ ∈ Top)) |
9 | 4, 8 | mtbiri 329 | . . . . 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 6673 | . . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
13 | 12, 5 | syl6eqr 2877 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
14 | fveq2 6673 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
15 | istps.a | . . . . . 6 ⊢ 𝐴 = (Base‘𝐾) | |
16 | 14, 15 | syl6eqr 2877 | . . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴) |
17 | 16 | fveq2d 6677 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴)) |
18 | 13, 17 | eleq12d 2910 | . . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴))) |
19 | 11, 18 | elab3 3677 | . 2 ⊢ (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴)) |
20 | 2, 19 | bitri 277 | 1 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1536 ∈ wcel 2113 {cab 2802 Vcvv 3497 ∅c0 4294 ‘cfv 6358 Basecbs 16486 TopOpenctopn 16698 Topctop 21504 TopOnctopon 21521 TopSpctps 21543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-top 21505 df-topon 21522 df-topsp 21544 |
This theorem is referenced by: istps2 21546 tpspropd 21549 tsettps 21552 indistps2ALT 21625 resstps 21798 prdstps 22240 imastps 22332 xpstopnlem2 22422 tmdtopon 22692 tgptopon 22693 istgp2 22702 oppgtmd 22708 distgp 22710 indistgp 22711 efmndtmd 22712 qustgplem 22732 prdstmdd 22735 eltsms 22744 tsmscls 22749 tsmsgsum 22750 tsmsid 22751 tsmsmhm 22757 tsmsadd 22758 dvrcn 22795 cnmpt1vsca 22805 cnmpt2vsca 22806 tlmtgp 22807 ressusp 22877 tustps 22885 ucncn 22897 neipcfilu 22908 cnextucn 22915 ucnextcn 22916 isxms2 23061 ressxms 23138 prdsxmslem2 23142 nrgtrg 23302 cnfldtopon 23394 cnmpt1ds 23453 cnmpt2ds 23454 nmcn 23455 cnmpt1ip 23853 cnmpt2ip 23854 csscld 23855 clsocv 23856 minveclem4a 24036 mhmhmeotmd 31174 rrxtopon 42580 qndenserrnopnlem 42589 |
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