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Theorem hmeof1o2 14191
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 14188 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
2 cnf2 14088 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
31, 2syl3an3 1283 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  F : X --> Y )
43ffnd 5380 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  F  Fn  X
)
5 hmeocnvcn 14189 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
6 cnf2 14088 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  J  e.  (TopOn `  X )  /\  `' F  e.  ( K  Cn  J ) )  ->  `' F : Y
--> X )
763com12 1208 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  `' F  e.  ( K  Cn  J ) )  ->  `' F : Y
--> X )
85, 7syl3an3 1283 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  `' F : Y
--> X )
98ffnd 5380 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  `' F  Fn  Y )
10 dff1o4 5483 . 2  |-  ( F : X -1-1-onto-> Y  <->  ( F  Fn  X  /\  `' F  Fn  Y ) )
114, 9, 10sylanbrc 417 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 979    e. wcel 2159   `'ccnv 4639    Fn wfn 5225   -->wf 5226   -1-1-onto->wf1o 5229   ` cfv 5230  (class class class)co 5890  TopOnctopon 13893    Cn ccn 14068   Homeochmeo 14183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-map 6667  df-top 13881  df-topon 13894  df-cn 14071  df-hmeo 14184
This theorem is referenced by:  hmeof1o  14192
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