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Theorem hmeof1o2 17454
Description: A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmeof1o2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Homeo  K ) )  ->  F : X -1-1-onto-> Y
)

Proof of Theorem hmeof1o2
StepHypRef Expression
1 hmeocn 17451 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
2 cnf2 16979 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
31, 2syl3an3 1217 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Homeo  K ) )  ->  F : X --> Y )
4 ffn 5389 . . 3  |-  ( F : X --> Y  ->  F  Fn  X )
53, 4syl 15 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Homeo  K ) )  ->  F  Fn  X
)
6 hmeocnvcn 17452 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
7 cnf2 16979 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  J  e.  (TopOn `  X )  /\  `' F  e.  ( K  Cn  J ) )  ->  `' F : Y
--> X )
873com12 1155 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  `' F  e.  ( K  Cn  J ) )  ->  `' F : Y
--> X )
96, 8syl3an3 1217 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Homeo  K ) )  ->  `' F : Y
--> X )
10 ffn 5389 . . 3  |-  ( `' F : Y --> X  ->  `' F  Fn  Y
)
119, 10syl 15 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Homeo  K ) )  ->  `' F  Fn  Y )
12 dff1o4 5480 . 2  |-  ( F : X -1-1-onto-> Y  <->  ( F  Fn  X  /\  `' F  Fn  Y ) )
135, 11, 12sylanbrc 645 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Homeo  K ) )  ->  F : X -1-1-onto-> Y
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684   `'ccnv 4688    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632    Cn ccn 16954    Homeo chmeo 17444
This theorem is referenced by:  hmeof1o  17455  qtophmeo  17508  cvmsf1o  23803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446
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