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Mirrors > Home > MPE Home > Th. List > hmeoqtop | Structured version Visualization version GIF version |
Description: A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
hmeoqtop | ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocn 23752 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | cntop2 23233 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top) |
4 | toptopon2 22908 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
5 | 3, 4 | sylib 217 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
6 | eqid 2726 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
7 | eqid 2726 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
8 | 6, 7 | hmeof1o 23756 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾) |
9 | f1ofo 6842 | . . 3 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–onto→∪ 𝐾) | |
10 | forn 6810 | . . 3 ⊢ (𝐹:∪ 𝐽–onto→∪ 𝐾 → ran 𝐹 = ∪ 𝐾) | |
11 | 8, 9, 10 | 3syl 18 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ran 𝐹 = ∪ 𝐾) |
12 | hmeoima 23757 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | |
13 | 5, 1, 11, 12 | qtopomap 23710 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∪ cuni 4905 ran crn 5675 –onto→wfo 6544 –1-1-onto→wf1o 6545 ‘cfv 6546 (class class class)co 7416 qTop cqtop 17513 Topctop 22883 TopOnctopon 22900 Cn ccn 23216 Homeochmeo 23745 |
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-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
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-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-map 8849 df-qtop 17517 df-top 22884 df-topon 22901 df-cn 23219 df-hmeo 23747 |
This theorem is referenced by: xpstopnlem2 23803 |
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