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Theorem hmeof1o2 20027
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 20024 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
2 cnf2 19544 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> Y )
31, 2syl3an3 1263 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  F : X --> Y )
4 ffn 5731 . . 3  |-  ( F : X --> Y  ->  F  Fn  X )
53, 4syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  F  Fn  X
)
6 hmeocnvcn 20025 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
7 cnf2 19544 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  J  e.  (TopOn `  X )  /\  `' F  e.  ( K  Cn  J ) )  ->  `' F : Y
--> X )
873com12 1200 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  `' F  e.  ( K  Cn  J ) )  ->  `' F : Y
--> X )
96, 8syl3an3 1263 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  `' F : Y
--> X )
10 ffn 5731 . . 3  |-  ( `' F : Y --> X  ->  `' F  Fn  Y
)
119, 10syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  `' F  Fn  Y )
12 dff1o4 5824 . 2  |-  ( F : X -1-1-onto-> Y  <->  ( F  Fn  X  /\  `' F  Fn  Y ) )
135, 11, 12sylanbrc 664 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J Homeo K ) )  ->  F : X -1-1-onto-> Y
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    e. wcel 1767   `'ccnv 4998    Fn wfn 5583   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284  TopOnctopon 19190    Cn ccn 19519   Homeochmeo 20017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-top 19194  df-topon 19197  df-cn 19522  df-hmeo 20019
This theorem is referenced by:  hmeof1o  20028  qtophmeo  20081  cvmsf1o  28385
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