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Theorem ishmeo 20705
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 5028 . . 3  |-  ( f  =  F  ->  `' f  =  `' F
)
21eleq1d 2498 . 2  |-  ( f  =  F  ->  ( `' f  e.  ( K  Cn  J )  <->  `' F  e.  ( K  Cn  J
) ) )
3 hmeofval 20704 . 2  |-  ( J
Homeo K )  =  {
f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }
42, 3elrab2 3237 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   `'ccnv 4853  (class class class)co 6305    Cn ccn 20171   Homeochmeo 20699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-top 19852  df-topon 19854  df-cn 20174  df-hmeo 20701
This theorem is referenced by:  hmeocn  20706  hmeocnvcn  20707  hmeocnv  20708  hmeores  20717  hmeoco  20718  idhmeo  20719  indishmph  20744  cmphaushmeo  20746  ordthmeo  20748  txhmeo  20749  txswaphmeo  20751  pt1hmeo  20752  ptunhmeo  20754  xkohmeo  20761  qtopf1  20762  qtophmeo  20763  grpinvhmeo  21032  tgplacthmeo  21049  cncfcnvcn  21849  icchmeo  21865  cnrehmeo  21877  cnheiborlem  21878  ismtyhmeo  31841
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