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Theorem hmeoimaf1o 17792
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 17787 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  x  e.  J )  ->  ( F " x )  e.  K )
3 hmeocn 17782 . . 3  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
4 cnima 17319 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  y  e.  K )  ->  ( `' F "
y )  e.  J
)
53, 4sylan 458 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  y  e.  K )  ->  ( `' F " y )  e.  J )
6 eqid 2435 . . . . . . 7  |-  U. J  =  U. J
7 eqid 2435 . . . . . . 7  |-  U. K  =  U. K
86, 7hmeof1o 17786 . . . . . 6  |-  ( F  e.  ( J  Homeo  K )  ->  F : U. J -1-1-onto-> U. K )
98adantr 452 . . . . 5  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  F : U. J -1-1-onto-> U. K )
10 f1of1 5665 . . . . 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 4035 . . . . 5  |-  ( x  e.  J  ->  x  C_ 
U. J )
1312ad2antrl 709 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  x  C_  U. J
)
14 cnvimass 5216 . . . . 5  |-  ( `' F " y ) 
C_  dom  F
15 f1dm 5635 . . . . . 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 3388 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( `' F " y )  C_  U. J )
18 f1imaeq 6003 . . . 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 1182 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( ( F " x )  =  ( F " ( `' F " y ) )  <->  x  =  ( `' F " y ) ) )
20 f1ofo 5673 . . . . . . 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 4035 . . . . . . 7  |-  ( y  e.  K  ->  y  C_ 
U. K )
2322ad2antll 710 . . . . . 6  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  y  C_  U. K )
24 foimacnv 5684 . . . . . 6  |-  ( ( F : U. J -onto-> U. K  /\  y  C_ 
U. K )  -> 
( F " ( `' F " y ) )  =  y )
2521, 23, 24syl2anc 643 . . . . 5  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( F " ( `' F "
y ) )  =  y )
2625eqeq2d 2446 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( ( F " x )  =  ( F " ( `' F " y ) )  <->  ( F "
x )  =  y ) )
27 eqcom 2437 . . . 4  |-  ( ( F " x )  =  y  <->  y  =  ( F " x ) )
2826, 27syl6bb 253 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( ( F " x )  =  ( F " ( `' F " y ) )  <->  y  =  ( F " x ) ) )
2919, 28bitr3d 247 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  (
x  e.  J  /\  y  e.  K )
)  ->  ( x  =  ( `' F " y )  <->  y  =  ( F " x ) ) )
301, 2, 5, 29f1o2d 6288 1  |-  ( F  e.  ( J  Homeo  K )  ->  G : J
-1-1-onto-> K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   U.cuni 4007    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   "cima 4873   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445  (class class class)co 6073    Cn ccn 17278    Homeo chmeo 17775
This theorem is referenced by:  hmphen  17807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-top 16953  df-topon 16956  df-cn 17281  df-hmeo 17777
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