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Theorem hmeofval 20553
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 6289 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( j  Cn  k
)  =  ( J  Cn  K ) )
2 oveq12 6289 . . . . . 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 2474 . . . 4  |-  ( ( j  =  J  /\  k  =  K )  ->  ( `' f  e.  ( k  Cn  j
)  <->  `' f  e.  ( K  Cn  J ) ) )
51, 4rabeqbidv 3056 . . 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 20550 . . 3  |-  Homeo  =  ( j  e.  Top , 
k  e.  Top  |->  { f  e.  ( j  Cn  k )  |  `' f  e.  (
k  Cn  j ) } )
7 ovex 6308 . . . 4  |-  ( J  Cn  K )  e. 
_V
87rabex 4547 . . 3  |-  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  e.  _V
95, 6, 8ovmpt2a 6416 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
106mpt2ndm0 6499 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  (/) )
11 cntop1 20036 . . . . . . . 8  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
12 cntop2 20037 . . . . . . . 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 26 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  ( `' f  e.  ( K  Cn  J )  -> 
( J  e.  Top  /\  K  e.  Top )
) )
1514con3rr3 138 . . . . 5  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( f  e.  ( J  Cn  K )  ->  -.  `' f  e.  ( K  Cn  J
) ) )
1615ralrimiv 2818 . . . 4  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
17 rabeq0 3763 . . . 4  |-  ( { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J ) }  =  (/)  <->  A. f  e.  ( J  Cn  K )  -.  `' f  e.  ( K  Cn  J
) )
1816, 17sylibr 214 . . 3  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  { f  e.  ( J  Cn  K )  |  `' f  e.  ( K  Cn  J
) }  =  (/) )
1910, 18eqtr4d 2448 . 2  |-  ( -.  ( J  e.  Top  /\  K  e.  Top )  ->  ( J Homeo K )  =  { f  e.  ( J  Cn  K
)  |  `' f  e.  ( K  Cn  J ) } )
209, 19pm2.61i 166 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 1407    e. wcel 1844   A.wral 2756   {crab 2760   (/)c0 3740   `'ccnv 4824  (class class class)co 6280   Topctop 19688    Cn ccn 20020   Homeochmeo 20548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-map 7461  df-top 19693  df-topon 19696  df-cn 20023  df-hmeo 20550
This theorem is referenced by:  ishmeo  20554
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