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Mirrors > Home > MPE Home > Th. List > hmeof1o | Structured version Visualization version GIF version |
Description: A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.) |
Ref | Expression |
---|---|
hmeof1o.1 | ⊢ 𝑋 = ∪ 𝐽 |
hmeof1o.2 | ⊢ 𝑌 = ∪ 𝐾 |
Ref | Expression |
---|---|
hmeof1o | ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocn 21976 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | cntop1 21456 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
3 | hmeof1o.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | toptopon 21133 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | 2, 4 | sylib 210 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | cntop2 21457 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
7 | hmeof1o.2 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
8 | 7 | toptopon 21133 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
9 | 6, 8 | sylib 210 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌)) |
10 | 5, 9 | jca 507 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌))) |
11 | 1, 10 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌))) |
12 | hmeof1o2 21979 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋–1-1-onto→𝑌) | |
13 | 12 | 3expia 1111 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)) |
14 | 11, 13 | mpcom 38 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∪ cuni 4673 –1-1-onto→wf1o 6136 ‘cfv 6137 (class class class)co 6924 Topctop 21109 TopOnctopon 21126 Cn ccn 21440 Homeochmeo 21969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-map 8144 df-top 21110 df-topon 21127 df-cn 21443 df-hmeo 21971 |
This theorem is referenced by: hmeoopn 21982 hmeocld 21983 hmeontr 21985 hmeoimaf1o 21986 hmeoqtop 21991 haushmphlem 22003 cmphmph 22004 connhmph 22005 reghmph 22009 nrmhmph 22010 hmphdis 22012 hmphen2 22015 cmphaushmeo 22016 txhmeo 22019 tpr2rico 30560 mndpluscn 30574 |
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