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Theorem hmeof1o 23627
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 23623 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2 cntop1 23103 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
3 hmeof1o.1 . . . . . 6 𝑋 = 𝐽
43toptopon 22780 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
52, 4sylib 218 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
6 cntop2 23104 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
7 hmeof1o.2 . . . . . 6 𝑌 = 𝐾
87toptopon 22780 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
96, 8sylib 218 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
105, 9jca 511 . . 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 23626 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋1-1-onto𝑌)
13123expia 1121 . 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 395   = wceq 1540  wcel 2109   cuni 4867  1-1-ontowf1o 6498  cfv 6499  (class class class)co 7369  Topctop 22756  TopOnctopon 22773   Cn ccn 23087  Homeochmeo 23616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-top 22757  df-topon 22774  df-cn 23090  df-hmeo 23618
This theorem is referenced by:  hmeoopn  23629  hmeocld  23630  hmeontr  23632  hmeoimaf1o  23633  hmeoqtop  23638  haushmphlem  23650  cmphmph  23651  connhmph  23652  reghmph  23656  nrmhmph  23657  hmphdis  23659  hmphen2  23662  cmphaushmeo  23663  txhmeo  23666  tpr2rico  33875  mndpluscn  33889
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