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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 23111 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | cntop2 22592 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top) |
4 | toptopon2 22267 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
5 | 3, 4 | sylib 217 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
6 | eqid 2736 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
7 | eqid 2736 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
8 | 6, 7 | hmeof1o 23115 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾) |
9 | f1ofo 6791 | . . 3 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–onto→∪ 𝐾) | |
10 | forn 6759 | . . 3 ⊢ (𝐹:∪ 𝐽–onto→∪ 𝐾 → ran 𝐹 = ∪ 𝐾) | |
11 | 8, 9, 10 | 3syl 18 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ran 𝐹 = ∪ 𝐾) |
12 | hmeoima 23116 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | |
13 | 5, 1, 11, 12 | qtopomap 23069 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∪ cuni 4865 ran crn 5634 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 qTop cqtop 17385 Topctop 22242 TopOnctopon 22259 Cn ccn 22575 Homeochmeo 23104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-map 8767 df-qtop 17389 df-top 22243 df-topon 22260 df-cn 22578 df-hmeo 23106 |
This theorem is referenced by: xpstopnlem2 23162 |
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