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Theorem hmeofval 17465
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 5883 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( j  Cn  k
)  =  ( J  Cn  K ) )
2 oveq12 5883 . . . . . 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 2363 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( `' f  e.  ( k  Cn  j
)  <->  `' f  e.  ( K  Cn  J ) ) )
51, 4rabeqbidv 2796 . . 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 17462 . . 3  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
7 ovex 5899 . . . 4  |-  ( J  Cn  K )  e. 
_V
87rabex 4181 . . 3  |-  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  e.  _V
95, 6, 8ovmpt2a 5994 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Homeo  K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
10 ovex 5899 . . . . . 6  |-  ( j  Cn  k )  e. 
_V
1110rabex 4181 . . . . 5  |-  { f  e.  ( j  Cn  k )  |  `' f  e.  ( k  Cn  j ) }  e.  _V
126, 11dmmpt2 6210 . . . 4  |-  dom  Homeo  =  ( Top  X.  Top )
1312ndmov 6020 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Homeo  K )  =  (/) )
14 cntop1 16986 . . . . . . . 8  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
15 cntop2 16987 . . . . . . . 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 2638 . . . 4  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
20 rabeq0 3489 . . . 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 2331 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   (/)c0 3468   `'ccnv 4704  (class class class)co 5874   Topctop 16647    Cn ccn 16970    Homeo chmeo 17460
This theorem is referenced by:  ishmeo  17466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-hmeo 17462
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