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Theorem ishmeo 19331
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 variable  f is distinct from all other variables.
StepHypRef Expression
1 cnveq 5012 . . 3  |-  ( f  =  F  ->  `' f  =  `' F
)
21eleq1d 2508 . 2  |-  ( f  =  F  ->  ( `' f  e.  ( K  Cn  J )  <->  `' F  e.  ( K  Cn  J
) ) )
3 hmeofval 19330 . 2  |-  ( J
Homeo K )  =  {
f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }
42, 3elrab2 3118 1  |-  ( F  e.  ( J Homeo K )  <->  ( F  e.  ( J  Cn  K
)  /\  `' F  e.  ( K  Cn  J
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   `'ccnv 4838  (class class class)co 6090    Cn ccn 18827   Homeochmeo 19325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7215  df-top 18502  df-topon 18505  df-cn 18830  df-hmeo 19327
This theorem is referenced by:  hmeocn  19332  hmeocnvcn  19333  hmeocnv  19334  hmeores  19343  hmeoco  19344  idhmeo  19345  indishmph  19370  cmphaushmeo  19372  ordthmeo  19374  txhmeo  19375  txswaphmeo  19377  pt1hmeo  19378  ptunhmeo  19380  xkohmeo  19387  qtopf1  19388  qtophmeo  19389  grpinvhmeo  19656  tgplacthmeo  19673  cncfcnvcn  20496  icchmeo  20512  cnrehmeo  20524  cnheiborlem  20525  ismtyhmeo  28702
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