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Theorem cnfval 16963
Description: The set of all continuous functions from topology  J to topology  K. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnfval  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  Cn  K )  =  {
f  e.  ( Y  ^m  X )  | 
A. y  e.  K  ( `' f " y
)  e.  J }
)
Distinct variable groups:    y, f, K    f, X, y    f, Y, y    f, J, y

Proof of Theorem cnfval
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cn 16957 . . 3  |-  Cn  =  ( j  e.  Top ,  k  e.  Top  |->  { f  e.  ( U. k  ^m  U. j )  |  A. y  e.  k  ( `' f
" y )  e.  j } )
21a1i 10 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  Cn  =  ( j  e.  Top ,  k  e.  Top  |->  { f  e.  ( U. k  ^m  U. j )  |  A. y  e.  k  ( `' f
" y )  e.  j } ) )
3 simprr 733 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
k  =  K )
43unieqd 3838 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. k  =  U. K )
5 toponuni 16665 . . . . . 6  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 707 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  Y  =  U. K )
74, 6eqtr4d 2318 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. k  =  Y
)
8 simprl 732 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
j  =  J )
98unieqd 3838 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  U. J )
10 toponuni 16665 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1110ad2antrr 706 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  X  =  U. J )
129, 11eqtr4d 2318 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  X
)
137, 12oveq12d 5876 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
( U. k  ^m  U. j )  =  ( Y  ^m  X ) )
148eleq2d 2350 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
( ( `' f
" y )  e.  j  <->  ( `' f
" y )  e.  J ) )
153, 14raleqbidv 2748 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
( A. y  e.  k  ( `' f
" y )  e.  j  <->  A. y  e.  K  ( `' f " y
)  e.  J ) )
1613, 15rabeqbidv 2783 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  { f  e.  ( U. k  ^m  U. j )  |  A. y  e.  k  ( `' f " y
)  e.  j }  =  { f  e.  ( Y  ^m  X
)  |  A. y  e.  K  ( `' f " y )  e.  J } )
17 topontop 16664 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1817adantr 451 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  J  e.  Top )
19 topontop 16664 . . 3  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
2019adantl 452 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  K  e.  Top )
21 ovex 5883 . . . 4  |-  ( Y  ^m  X )  e. 
_V
2221rabex 4165 . . 3  |-  { f  e.  ( Y  ^m  X )  |  A. y  e.  K  ( `' f " y
)  e.  J }  e.  _V
2322a1i 10 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  { f  e.  ( Y  ^m  X
)  |  A. y  e.  K  ( `' f " y )  e.  J }  e.  _V )
242, 16, 18, 20, 23ovmpt2d 5975 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  Cn  K )  =  {
f  e.  ( Y  ^m  X )  | 
A. y  e.  K  ( `' f " y
)  e.  J }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   U.cuni 3827   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ^m cmap 6772   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  iscn  16965  cnfex  27699
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-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-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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-topon 16639  df-cn 16957
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