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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. Proposition 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 5328 | . . 3 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
2 | 1 | eleq1d 2715 | . 2 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝐾 Cn 𝐽) ↔ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
3 | hmeofval 21609 | . 2 ⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} | |
4 | 2, 3 | elrab2 3399 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ◡ccnv 5142 (class class class)co 6690 Cn ccn 21076 Homeochmeo 21604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-map 7901 df-top 20747 df-topon 20764 df-cn 21079 df-hmeo 21606 |
This theorem is referenced by: hmeocn 21611 hmeocnvcn 21612 hmeocnv 21613 hmeores 21622 hmeoco 21623 idhmeo 21624 indishmph 21649 cmphaushmeo 21651 ordthmeo 21653 txhmeo 21654 txswaphmeo 21656 pt1hmeo 21657 ptunhmeo 21659 xkohmeo 21666 qtopf1 21667 qtophmeo 21668 grpinvhmeo 21937 tgplacthmeo 21954 cncfcnvcn 22771 icchmeo 22787 cnrehmeo 22799 cnheiborlem 22800 ismtyhmeo 33734 |
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