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Theorem hmeofval 17790
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 6090 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( j  Cn  k
)  =  ( J  Cn  K ) )
2 oveq12 6090 . . . . . 6  |-  ( ( k  =  K  /\  j  =  J )  ->  ( k  Cn  j
)  =  ( K  Cn  J ) )
32ancoms 440 . . . . 5  |-  ( ( j  =  J  /\  k  =  K )  ->  ( k  Cn  j
)  =  ( K  Cn  J ) )
43eleq2d 2503 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( `' f  e.  ( k  Cn  j
)  <->  `' f  e.  ( K  Cn  J ) ) )
51, 4rabeqbidv 2951 . . 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 17787 . . 3  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
7 ovex 6106 . . . 4  |-  ( J  Cn  K )  e. 
_V
87rabex 4354 . . 3  |-  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  e.  _V
95, 6, 8ovmpt2a 6204 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Homeo  K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
106mpt2ndm0 6473 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Homeo  K )  =  (/) )
11 cntop1 17304 . . . . . . . 8  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
12 cntop2 17305 . . . . . . . 8  |-  ( f  e.  ( J  Cn  K )  ->  K  e.  Top )
1311, 12jca 519 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  ( J  e.  Top  /\  K  e.  Top ) )
1413a1d 23 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  ( `' f  e.  ( K  Cn  J )  -> 
( J  e.  Top  /\  K  e.  Top )
) )
1514con3rr3 130 . . . . 5  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( f  e.  ( J  Cn  K )  ->  -.  `' f  e.  ( K  Cn  J
) ) )
1615ralrimiv 2788 . . . 4  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
17 rabeq0 3649 . . . 4  |-  ( { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  =  (/)  <->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
1816, 17sylibr 204 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J
) }  =  (/) )
1910, 18eqtr4d 2471 . 2  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Homeo  K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
209, 19pm2.61i 158 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 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   (/)c0 3628   `'ccnv 4877  (class class class)co 6081   Topctop 16958    Cn ccn 17288    Homeo chmeo 17785
This theorem is referenced by:  ishmeo  17791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-top 16963  df-topon 16966  df-cn 17291  df-hmeo 17787
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