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Theorem hmeofn 17712
Description: The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmeofn  |-  Homeo  Fn  ( Top  X.  Top )

Proof of Theorem hmeofn
Dummy variables  f 
j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hmeo 17710 . 2  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
2 ovex 6047 . . 3  |-  ( j  Cn  k )  e. 
_V
32rabex 4297 . 2  |-  { f  e.  ( j  Cn  k )  |  `' f  e.  ( k  Cn  j ) }  e.  _V
41, 3fnmpt2i 6361 1  |-  Homeo  Fn  ( Top  X.  Top )
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   {crab 2655    X. cxp 4818   `'ccnv 4819    Fn wfn 5391  (class class class)co 6022   Topctop 16883    Cn ccn 17212    Homeo chmeo 17708
This theorem is referenced by:  hmph  17731  hmphtop  17733  hmpher  17739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-hmeo 17710
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