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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 22370 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | cntop1 21850 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
3 | hmeof1o.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | toptopon 21527 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | 2, 4 | sylib 220 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | cntop2 21851 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
7 | hmeof1o.2 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
8 | 7 | toptopon 21527 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
9 | 6, 8 | sylib 220 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌)) |
10 | 5, 9 | jca 514 | . . 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 22373 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋–1-1-onto→𝑌) | |
13 | 12 | 3expia 1117 | . 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 398 = wceq 1537 ∈ wcel 2114 ∪ cuni 4840 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 Topctop 21503 TopOnctopon 21520 Cn ccn 21834 Homeochmeo 22363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-top 21504 df-topon 21521 df-cn 21837 df-hmeo 22365 |
This theorem is referenced by: hmeoopn 22376 hmeocld 22377 hmeontr 22379 hmeoimaf1o 22380 hmeoqtop 22385 haushmphlem 22397 cmphmph 22398 connhmph 22399 reghmph 22403 nrmhmph 22404 hmphdis 22406 hmphen2 22409 cmphaushmeo 22410 txhmeo 22413 tpr2rico 31157 mndpluscn 31171 |
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