![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 21941 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | cntop2 21423 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top) |
4 | eqid 2825 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 4 | toptopon 21099 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
6 | 3, 5 | sylib 210 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
7 | eqid 2825 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
8 | 7, 4 | hmeof1o 21945 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾) |
9 | f1ofo 6389 | . . 3 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–onto→∪ 𝐾) | |
10 | forn 6360 | . . 3 ⊢ (𝐹:∪ 𝐽–onto→∪ 𝐾 → ran 𝐹 = ∪ 𝐾) | |
11 | 8, 9, 10 | 3syl 18 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ran 𝐹 = ∪ 𝐾) |
12 | hmeoima 21946 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | |
13 | 6, 1, 11, 12 | qtopomap 21899 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ∪ cuni 4660 ran crn 5347 –onto→wfo 6125 –1-1-onto→wf1o 6126 ‘cfv 6127 (class class class)co 6910 qTop cqtop 16523 Topctop 21075 TopOnctopon 21092 Cn ccn 21406 Homeochmeo 21934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-qtop 16527 df-top 21076 df-topon 21093 df-cn 21409 df-hmeo 21936 |
This theorem is referenced by: xpstopnlem2 21992 |
Copyright terms: Public domain | W3C validator |