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Theorem ishmeo 12746
Description: The predicate F is a homeomorphism between topology  J and topology  K. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
ishmeo  |-  ( F  e.  ( J Homeo K )  <->  ( F  e.  ( J  Cn  K
)  /\  `' F  e.  ( K  Cn  J
) ) )

Proof of Theorem ishmeo
Dummy variables  f  j  k  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hmeo 12743 . . 3  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
21elmpocl 6019 . 2  |-  ( F  e.  ( J Homeo K )  ->  ( J  e.  Top  /\  K  e. 
Top ) )
3 df-cn 12630 . . . 4  |-  Cn  =  ( j  e.  Top ,  k  e.  Top  |->  { f  e.  ( U. k  ^m  U. j )  |  A. y  e.  k  ( `' f
" y )  e.  j } )
43elmpocl 6019 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  ( J  e.  Top  /\  K  e.  Top ) )
54adantr 274 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  `' F  e.  ( K  Cn  J ) )  ->  ( J  e. 
Top  /\  K  e.  Top ) )
6 hmeofvalg 12745 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
76eleq2d 2227 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( F  e.  ( J Homeo K )  <->  F  e.  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) } ) )
8 cnveq 4761 . . . . 5  |-  ( f  =  F  ->  `' f  =  `' F
)
98eleq1d 2226 . . . 4  |-  ( f  =  F  ->  ( `' f  e.  ( K  Cn  J )  <->  `' F  e.  ( K  Cn  J
) ) )
109elrab 2868 . . 3  |-  ( F  e.  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) }  <->  ( F  e.  ( J  Cn  K
)  /\  `' F  e.  ( K  Cn  J
) ) )
117, 10bitrdi 195 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( F  e.  ( J Homeo K )  <->  ( F  e.  ( J  Cn  K
)  /\  `' F  e.  ( K  Cn  J
) ) ) )
122, 5, 11pm5.21nii 694 1  |-  ( F  e.  ( J Homeo K )  <->  ( F  e.  ( J  Cn  K
)  /\  `' F  e.  ( K  Cn  J
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   A.wral 2435   {crab 2439   U.cuni 3773   `'ccnv 4586   "cima 4590  (class class class)co 5825    ^m cmap 6594   Topctop 12437    Cn ccn 12627   Homeochmeo 12742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-map 6596  df-top 12438  df-topon 12451  df-cn 12630  df-hmeo 12743
This theorem is referenced by:  hmeocn  12747  hmeocnvcn  12748  hmeocnv  12749  hmeores  12757  hmeoco  12758  idhmeo  12759  txhmeo  12761  txswaphmeo  12763  cnrehmeocntop  13035
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