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Theorem hmeofval 19989
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 6286 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( j  Cn  k
)  =  ( J  Cn  K ) )
2 oveq12 6286 . . . . . 6  |-  ( ( k  =  K  /\  j  =  J )  ->  ( k  Cn  j
)  =  ( K  Cn  J ) )
32ancoms 453 . . . . 5  |-  ( ( j  =  J  /\  k  =  K )  ->  ( k  Cn  j
)  =  ( K  Cn  J ) )
43eleq2d 2532 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( `' f  e.  ( k  Cn  j
)  <->  `' f  e.  ( K  Cn  J ) ) )
51, 4rabeqbidv 3103 . . 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 19986 . . 3  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
7 ovex 6302 . . . 4  |-  ( J  Cn  K )  e. 
_V
87rabex 4593 . . 3  |-  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  e.  _V
95, 6, 8ovmpt2a 6410 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
106mpt2ndm0 6493 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  (/) )
11 cntop1 19502 . . . . . . . 8  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
12 cntop2 19503 . . . . . . . 8  |-  ( f  e.  ( J  Cn  K )  ->  K  e.  Top )
1311, 12jca 532 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  ( J  e.  Top  /\  K  e.  Top ) )
1413a1d 25 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  ( `' f  e.  ( K  Cn  J )  -> 
( J  e.  Top  /\  K  e.  Top )
) )
1514con3rr3 136 . . . . 5  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( f  e.  ( J  Cn  K )  ->  -.  `' f  e.  ( K  Cn  J
) ) )
1615ralrimiv 2871 . . . 4  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
17 rabeq0 3802 . . . 4  |-  ( { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  =  (/)  <->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
1816, 17sylibr 212 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J
) }  =  (/) )
1910, 18eqtr4d 2506 . 2  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
209, 19pm2.61i 164 1  |-  ( J
Homeo K )  =  {
f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   {crab 2813   (/)c0 3780   `'ccnv 4993  (class class class)co 6277   Topctop 19156    Cn ccn 19486   Homeochmeo 19984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-top 19161  df-topon 19164  df-cn 19489  df-hmeo 19986
This theorem is referenced by:  ishmeo  19990
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