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Theorem hmeoimaf1o 20397
Description: The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
hmeoimaf1o.1  |-  G  =  ( x  e.  J  |->  ( F " x
) )
Assertion
Ref Expression
hmeoimaf1o  |-  ( F  e.  ( J Homeo K )  ->  G : J
-1-1-onto-> K )
Distinct variable groups:    x, F    x, J    x, K
Allowed substitution hint:    G( x)

Proof of Theorem hmeoimaf1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 hmeoimaf1o.1 . 2  |-  G  =  ( x  e.  J  |->  ( F " x
) )
2 hmeoima 20392 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  x  e.  J )  ->  ( F " x )  e.  K )
3 hmeocn 20387 . . 3  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
4 cnima 19893 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  y  e.  K )  ->  ( `' F "
y )  e.  J
)
53, 4sylan 471 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  y  e.  K )  ->  ( `' F " y )  e.  J )
6 eqid 2457 . . . . . . 7  |-  U. J  =  U. J
7 eqid 2457 . . . . . . 7  |-  U. K  =  U. K
86, 7hmeof1o 20391 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : U. J -1-1-onto-> U. K )
98adantr 465 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  F : U. J -1-1-onto-> U. K )
10 f1of1 5821 . . . . 5  |-  ( F : U. J -1-1-onto-> U. K  ->  F : U. J -1-1-> U. K )
119, 10syl 16 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  F : U. J -1-1-> U. K )
12 elssuni 4281 . . . . 5  |-  ( x  e.  J  ->  x  C_ 
U. J )
1312ad2antrl 727 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  x  C_  U. J
)
14 cnvimass 5367 . . . . 5  |-  ( `' F " y ) 
C_  dom  F
15 f1dm 5791 . . . . . 6  |-  ( F : U. J -1-1-> U. K  ->  dom  F  =  U. J )
1611, 15syl 16 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  dom  F  = 
U. J )
1714, 16syl5sseq 3547 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( `' F " y )  C_  U. J )
18 f1imaeq 6174 . . . 4  |-  ( ( F : U. J -1-1-> U. K  /\  ( x 
C_  U. J  /\  ( `' F " y ) 
C_  U. J ) )  ->  ( ( F
" x )  =  ( F " ( `' F " y ) )  <->  x  =  ( `' F " y ) ) )
1911, 13, 17, 18syl12anc 1226 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( ( F " x )  =  ( F " ( `' F " y ) )  <->  x  =  ( `' F " y ) ) )
20 f1ofo 5829 . . . . . . 7  |-  ( F : U. J -1-1-onto-> U. K  ->  F : U. J -onto-> U. K )
219, 20syl 16 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  F : U. J -onto-> U. K )
22 elssuni 4281 . . . . . . 7  |-  ( y  e.  K  ->  y  C_ 
U. K )
2322ad2antll 728 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  y  C_  U. K )
24 foimacnv 5839 . . . . . 6  |-  ( ( F : U. J -onto-> U. K  /\  y  C_ 
U. K )  -> 
( F " ( `' F " y ) )  =  y )
2521, 23, 24syl2anc 661 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( F " ( `' F "
y ) )  =  y )
2625eqeq2d 2471 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( ( F " x )  =  ( F " ( `' F " y ) )  <->  ( F "
x )  =  y ) )
27 eqcom 2466 . . . 4  |-  ( ( F " x )  =  y  <->  y  =  ( F " x ) )
2826, 27syl6bb 261 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( ( F " x )  =  ( F " ( `' F " y ) )  <->  y  =  ( F " x ) ) )
2919, 28bitr3d 255 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( x  =  ( `' F " y )  <->  y  =  ( F " x ) ) )
301, 2, 5, 29f1o2d 6526 1  |-  ( F  e.  ( J Homeo K )  ->  G : J
-1-1-onto-> K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    C_ wss 3471   U.cuni 4251    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   "cima 5011   -1-1->wf1 5591   -onto->wfo 5592   -1-1-onto->wf1o 5593  (class class class)co 6296    Cn ccn 19852   Homeochmeo 20380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-top 19526  df-topon 19529  df-cn 19855  df-hmeo 20382
This theorem is referenced by:  hmphen  20412
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