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Theorem cnfval 16979
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 16973 . . 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 3854 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. k  =  U. K )
5 toponuni 16681 . . . . . 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 2331 . . . 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 3854 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  U. J )
10 toponuni 16681 . . . . . 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 2331 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  X
)
137, 12oveq12d 5892 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
( U. k  ^m  U. j )  =  ( Y  ^m  X ) )
148eleq2d 2363 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
( ( `' f
" y )  e.  j  <->  ( `' f
" y )  e.  J ) )
153, 14raleqbidv 2761 . . 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 2796 . 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 16680 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1817adantr 451 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  J  e.  Top )
19 topontop 16680 . . 3  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
2019adantl 452 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  K  e.  Top )
21 ovex 5899 . . . 4  |-  ( Y  ^m  X )  e. 
_V
2221rabex 4181 . . 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 5991 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   U.cuni 3843   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^m cmap 6788   Topctop 16647  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  iscn  16981  cnfex  27802
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-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-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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-topon 16655  df-cn 16973
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