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Theorem hmeofval 17449
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
hmeofval  |-  ( J 
Homeo  K )  =  {
f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }
Distinct variable groups:    f, J    f, K

Proof of Theorem hmeofval
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 5867 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( j  Cn  k
)  =  ( J  Cn  K ) )
2 oveq12 5867 . . . . . 6  |-  ( ( k  =  K  /\  j  =  J )  ->  ( k  Cn  j
)  =  ( K  Cn  J ) )
32ancoms 439 . . . . 5  |-  ( ( j  =  J  /\  k  =  K )  ->  ( k  Cn  j
)  =  ( K  Cn  J ) )
43eleq2d 2350 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( `' f  e.  ( k  Cn  j
)  <->  `' f  e.  ( K  Cn  J ) ) )
51, 4rabeqbidv 2783 . . 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 ) } )
6 df-hmeo 17446 . . 3  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
7 ovex 5883 . . . 4  |-  ( J  Cn  K )  e. 
_V
87rabex 4165 . . 3  |-  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  e.  _V
95, 6, 8ovmpt2a 5978 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Homeo  K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
10 ovex 5883 . . . . . 6  |-  ( j  Cn  k )  e. 
_V
1110rabex 4165 . . . . 5  |-  { f  e.  ( j  Cn  k )  |  `' f  e.  ( k  Cn  j ) }  e.  _V
126, 11dmmpt2 6194 . . . 4  |-  dom  Homeo  =  ( Top  X.  Top )
1312ndmov 6004 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Homeo  K )  =  (/) )
14 cntop1 16970 . . . . . . . 8  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
15 cntop2 16971 . . . . . . . 8  |-  ( f  e.  ( J  Cn  K )  ->  K  e.  Top )
1614, 15jca 518 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  ( J  e.  Top  /\  K  e.  Top ) )
1716a1d 22 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  ( `' f  e.  ( K  Cn  J )  -> 
( J  e.  Top  /\  K  e.  Top )
) )
1817con3rr3 128 . . . . 5  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( f  e.  ( J  Cn  K )  ->  -.  `' f  e.  ( K  Cn  J
) ) )
1918ralrimiv 2625 . . . 4  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
20 rabeq0 3476 . . . 4  |-  ( { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  =  (/)  <->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
2119, 20sylibr 203 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J
) }  =  (/) )
2213, 21eqtr4d 2318 . 2  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Homeo  K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
239, 22pm2.61i 156 1  |-  ( J 
Homeo  K )  =  {
f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   (/)c0 3455   `'ccnv 4688  (class class class)co 5858   Topctop 16631    Cn ccn 16954    Homeo chmeo 17444
This theorem is referenced by:  ishmeo  17450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446
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