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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 5737 | . . 3 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
2 | 1 | eleq1d 2894 | . 2 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝐾 Cn 𝐽) ↔ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
3 | hmeofval 22294 | . 2 ⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} | |
4 | 2, 3 | elrab2 3680 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ◡ccnv 5547 (class class class)co 7145 Cn ccn 21760 Homeochmeo 22289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-top 21430 df-topon 21447 df-cn 21763 df-hmeo 22291 |
This theorem is referenced by: hmeocn 22296 hmeocnvcn 22297 hmeocnv 22298 hmeores 22307 hmeoco 22308 idhmeo 22309 indishmph 22334 cmphaushmeo 22336 ordthmeo 22338 txhmeo 22339 txswaphmeo 22341 pt1hmeo 22342 ptunhmeo 22344 xkohmeo 22351 qtopf1 22352 qtophmeo 22353 grpinvhmeo 22622 tgplacthmeo 22639 cncfcnvcn 23456 icchmeo 23472 cnrehmeo 23484 cnheiborlem 23485 ismtyhmeo 34964 |
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