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Theorem hmeofvalg 13018
Description: The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmeofvalg  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
Distinct variable groups:    f, J    f, K

Proof of Theorem hmeofvalg
Dummy variables  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnovex 12911 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Cn  K
)  e.  _V )
2 rabexg 4130 . . 3  |-  ( ( J  Cn  K )  e.  _V  ->  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  e.  _V )
31, 2syl 14 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J
) }  e.  _V )
4 oveq12 5859 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( j  Cn  k
)  =  ( J  Cn  K ) )
5 oveq12 5859 . . . . . 6  |-  ( ( k  =  K  /\  j  =  J )  ->  ( k  Cn  j
)  =  ( K  Cn  J ) )
65ancoms 266 . . . . 5  |-  ( ( j  =  J  /\  k  =  K )  ->  ( k  Cn  j
)  =  ( K  Cn  J ) )
76eleq2d 2240 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( `' f  e.  ( k  Cn  j
)  <->  `' f  e.  ( K  Cn  J ) ) )
84, 7rabeqbidv 2725 . . 3  |-  ( ( j  =  J  /\  k  =  K )  ->  { f  e.  ( j  Cn  k )  |  `' f  e.  ( k  Cn  j
) }  =  {
f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) } )
9 df-hmeo 13016 . . 3  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
108, 9ovmpoga 5979 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  {
f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  e.  _V )  -> 
( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
113, 10mpd3an3 1333 1  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   {crab 2452   _Vcvv 2730   `'ccnv 4608  (class class class)co 5850   Topctop 12710    Cn ccn 12900   Homeochmeo 13015
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 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-map 6624  df-top 12711  df-topon 12724  df-cn 12903  df-hmeo 13016
This theorem is referenced by:  ishmeo  13019
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