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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 22296 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | cntop2 21777 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top) |
4 | toptopon2 21454 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
5 | 3, 4 | sylib 219 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
6 | eqid 2818 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
7 | eqid 2818 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
8 | 6, 7 | hmeof1o 22300 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾) |
9 | f1ofo 6615 | . . 3 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–onto→∪ 𝐾) | |
10 | forn 6586 | . . 3 ⊢ (𝐹:∪ 𝐽–onto→∪ 𝐾 → ran 𝐹 = ∪ 𝐾) | |
11 | 8, 9, 10 | 3syl 18 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ran 𝐹 = ∪ 𝐾) |
12 | hmeoima 22301 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | |
13 | 5, 1, 11, 12 | qtopomap 22254 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∪ cuni 4830 ran crn 5549 –onto→wfo 6346 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 qTop cqtop 16764 Topctop 21429 TopOnctopon 21446 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-rep 5181 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-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 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-iun 4912 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-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-qtop 16768 df-top 21430 df-topon 21447 df-cn 21763 df-hmeo 22291 |
This theorem is referenced by: xpstopnlem2 22347 |
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