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| Mirrors > Home > ILE Home > Th. List > ishmeo | 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 | df-hmeo 14773 | . . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)}) | |
| 2 | 1 | elmpocl 6141 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
| 3 | df-cn 14660 | . . . 4 ⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗}) | |
| 4 | 3 | elmpocl 6141 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
| 5 | 4 | adantr 276 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽)) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
| 6 | hmeofvalg 14775 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)}) | |
| 7 | 6 | eleq2d 2275 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})) |
| 8 | cnveq 4852 | . . . . 5 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
| 9 | 8 | eleq1d 2274 | . . . 4 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝐾 Cn 𝐽) ↔ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
| 10 | 9 | elrab 2929 | . . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
| 11 | 7, 10 | bitrdi 196 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽)))) |
| 12 | 2, 5, 11 | pm5.21nii 706 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
| Colors of variables: wff set class |
| Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1373 ∈ wcel 2176 ∀wral 2484 {crab 2488 ∪ cuni 3850 ◡ccnv 4674 “ cima 4678 (class class class)co 5944 ↑𝑚 cmap 6735 Topctop 14469 Cn ccn 14657 Homeochmeo 14772 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-map 6737 df-top 14470 df-topon 14483 df-cn 14660 df-hmeo 14773 |
| This theorem is referenced by: hmeocn 14777 hmeocnvcn 14778 hmeocnv 14779 hmeores 14787 hmeoco 14788 idhmeo 14789 txhmeo 14791 txswaphmeo 14793 cnrehmeocntop 15082 |
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