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Theorem ishmeo 13098
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 13095 . . 3  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
21elmpocl 6047 . 2  |-  ( F  e.  ( J Homeo K )  ->  ( J  e.  Top  /\  K  e. 
Top ) )
3 df-cn 12982 . . . 4  |-  Cn  =  ( j  e.  Top ,  k  e.  Top  |->  { f  e.  ( U. k  ^m  U. j )  |  A. y  e.  k  ( `' f
" y )  e.  j } )
43elmpocl 6047 . . 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 13097 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
76eleq2d 2240 . . 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 4785 . . . . 5  |-  ( f  =  F  ->  `' f  =  `' F
)
98eleq1d 2239 . . . 4  |-  ( f  =  F  ->  ( `' f  e.  ( K  Cn  J )  <->  `' F  e.  ( K  Cn  J
) ) )
109elrab 2886 . . 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 699 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 1348    e. wcel 2141   A.wral 2448   {crab 2452   U.cuni 3796   `'ccnv 4610   "cima 4614  (class class class)co 5853    ^m cmap 6626   Topctop 12789    Cn ccn 12979   Homeochmeo 13094
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-top 12790  df-topon 12803  df-cn 12982  df-hmeo 13095
This theorem is referenced by:  hmeocn  13099  hmeocnvcn  13100  hmeocnv  13101  hmeores  13109  hmeoco  13110  idhmeo  13111  txhmeo  13113  txswaphmeo  13115  cnrehmeocntop  13387
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