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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 20524 | . . 3 ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}) |
3 | topontop 20541 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
4 | 0ntop 20535 | . . . . . 6 ⊢ ¬ ∅ ∈ Top | |
5 | istps.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐾) | |
6 | fvprc 6097 | . . . . . . . 8 ⊢ (¬ 𝐾 ∈ V → (TopOpen‘𝐾) = ∅) | |
7 | 5, 6 | syl5eq 2656 | . . . . . . 7 ⊢ (¬ 𝐾 ∈ V → 𝐽 = ∅) |
8 | 7 | eleq1d 2672 | . . . . . 6 ⊢ (¬ 𝐾 ∈ V → (𝐽 ∈ Top ↔ ∅ ∈ Top)) |
9 | 4, 8 | mtbiri 316 | . . . . 5 ⊢ (¬ 𝐾 ∈ V → ¬ 𝐽 ∈ Top) |
10 | 9 | con4i 112 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ V) |
11 | 3, 10 | syl 17 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V) |
12 | fveq2 6103 | . . . . 5 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
13 | 12, 5 | syl6eqr 2662 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
14 | fveq2 6103 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
15 | istps.a | . . . . . 6 ⊢ 𝐴 = (Base‘𝐾) | |
16 | 14, 15 | syl6eqr 2662 | . . . . 5 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴) |
17 | 16 | fveq2d 6107 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴)) |
18 | 13, 17 | eleq12d 2682 | . . 3 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴))) |
19 | 11, 18 | elab3 3327 | . 2 ⊢ (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴)) |
20 | 2, 19 | bitri 263 | 1 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 ∅c0 3874 ‘cfv 5804 Basecbs 15695 TopOpenctopn 15905 Topctop 20517 TopOnctopon 20518 TopSpctps 20519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-top 20521 df-topon 20523 df-topsp 20524 |
This theorem is referenced by: istps2 20552 tpspropd 20555 tsettps 20558 indistps2ALT 20628 resstps 20801 prdstps 21242 imastps 21334 xpstopnlem2 21424 tmdtopon 21695 tgptopon 21696 istgp2 21705 oppgtmd 21711 distgp 21713 indistgp 21714 symgtgp 21715 qustgplem 21734 prdstmdd 21737 eltsms 21746 tsmscls 21751 tsmsgsum 21752 tsmsid 21753 tsmsmhm 21759 tsmsadd 21760 dvrcn 21797 cnmpt1vsca 21807 cnmpt2vsca 21808 tlmtgp 21809 ressusp 21879 tustps 21887 ucncn 21899 neipcfilu 21910 cnextucn 21917 ucnextcn 21918 isxms2 22063 ressxms 22140 prdsxmslem2 22144 nrgtrg 22304 cnfldtopon 22396 cnmpt1ds 22453 cnmpt2ds 22454 nmcn 22455 cnmpt1ip 22854 cnmpt2ip 22855 csscld 22856 clsocv 22857 minveclem4a 23009 mhmhmeotmd 29301 rrxtopon 39184 qndenserrnopnlem 39193 |
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