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Theorem hmeof1o 22300
Description: A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
Hypotheses
Ref Expression
hmeof1o.1 𝑋 = 𝐽
hmeof1o.2 𝑌 = 𝐾
Assertion
Ref Expression
hmeof1o (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto𝑌)

Proof of Theorem hmeof1o
StepHypRef Expression
1 hmeocn 22296 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2 cntop1 21776 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
3 hmeof1o.1 . . . . . 6 𝑋 = 𝐽
43toptopon 21453 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
52, 4sylib 219 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
6 cntop2 21777 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
7 hmeof1o.2 . . . . . 6 𝑌 = 𝐾
87toptopon 21453 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
96, 8sylib 219 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
105, 9jca 512 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)))
111, 10syl 17 . 2 (𝐹 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)))
12 hmeof1o2 22299 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋1-1-onto𝑌)
13123expia 1113 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto𝑌))
1411, 13mpcom 38 1 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105   cuni 4830  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  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-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-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  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-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-top 21430  df-topon 21447  df-cn 21763  df-hmeo 22291
This theorem is referenced by:  hmeoopn  22302  hmeocld  22303  hmeontr  22305  hmeoimaf1o  22306  hmeoqtop  22311  haushmphlem  22323  cmphmph  22324  connhmph  22325  reghmph  22329  nrmhmph  22330  hmphdis  22332  hmphen2  22335  cmphaushmeo  22336  txhmeo  22339  tpr2rico  31054  mndpluscn  31068
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