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Theorem ishmeo 20023
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 5176 . . 3  |-  ( f  =  F  ->  `' f  =  `' F
)
21eleq1d 2536 . 2  |-  ( f  =  F  ->  ( `' f  e.  ( K  Cn  J )  <->  `' F  e.  ( K  Cn  J
) ) )
3 hmeofval 20022 . 2  |-  ( J
Homeo K )  =  {
f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }
42, 3elrab2 3263 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 1379    e. wcel 1767   `'ccnv 4998  (class class class)co 6284    Cn ccn 19519   Homeochmeo 20017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-top 19194  df-topon 19197  df-cn 19522  df-hmeo 20019
This theorem is referenced by:  hmeocn  20024  hmeocnvcn  20025  hmeocnv  20026  hmeores  20035  hmeoco  20036  idhmeo  20037  indishmph  20062  cmphaushmeo  20064  ordthmeo  20066  txhmeo  20067  txswaphmeo  20069  pt1hmeo  20070  ptunhmeo  20072  xkohmeo  20079  qtopf1  20080  qtophmeo  20081  grpinvhmeo  20348  tgplacthmeo  20365  cncfcnvcn  21188  icchmeo  21204  cnrehmeo  21216  cnheiborlem  21217  ismtyhmeo  29932
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