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Theorem hmeof1o2 21787
Description: A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmeof1o2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋1-1-onto𝑌)

Proof of Theorem hmeof1o2
StepHypRef Expression
1 hmeocn 21784 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2 cnf2 21274 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
31, 2syl3an3 1169 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋𝑌)
43ffnd 6185 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹 Fn 𝑋)
5 hmeocnvcn 21785 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
6 cnf2 21274 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)) → 𝐹:𝑌𝑋)
763com12 1117 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)) → 𝐹:𝑌𝑋)
85, 7syl3an3 1169 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑌𝑋)
98ffnd 6185 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹 Fn 𝑌)
10 dff1o4 6287 . 2 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
114, 9, 10sylanbrc 572 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋1-1-onto𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071  wcel 2145  ccnv 5249   Fn wfn 6025  wf 6026  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  TopOnctopon 20935   Cn ccn 21249  Homeochmeo 21777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-top 20919  df-topon 20936  df-cn 21252  df-hmeo 21779
This theorem is referenced by:  hmeof1o  21788  qtophmeo  21841  cvmsf1o  31592
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