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Mirrors > Home > MPE Home > Th. List > ishmeo | Structured version Visualization version GIF version |
Description: The predicate F is a homeomorphism between topology 𝐽 and topology 𝐾. Criterion of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
ishmeo | ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5880 | . . 3 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
2 | 1 | eleq1d 2811 | . 2 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝐾 Cn 𝐽) ↔ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
3 | hmeofval 23753 | . 2 ⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} | |
4 | 2, 3 | elrab2 3684 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ◡ccnv 5681 (class class class)co 7424 Cn ccn 23219 Homeochmeo 23748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-map 8857 df-top 22887 df-topon 22904 df-cn 23222 df-hmeo 23750 |
This theorem is referenced by: hmeocn 23755 hmeocnvcn 23756 hmeocnv 23757 hmeores 23766 hmeoco 23767 idhmeo 23768 indishmph 23793 cmphaushmeo 23795 ordthmeo 23797 txhmeo 23798 txswaphmeo 23800 pt1hmeo 23801 ptunhmeo 23803 xkohmeo 23810 qtopf1 23811 qtophmeo 23812 grpinvhmeo 24081 tgplacthmeo 24098 cncfcnvcn 24937 icchmeo 24956 icchmeoOLD 24957 cnrehmeo 24969 cnrehmeoOLD 24970 cnheiborlem 24971 ismtyhmeo 37506 |
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