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Theorem cnfval 17299
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 17293 . . 3  |-  Cn  =  ( j  e.  Top ,  k  e.  Top  |->  { f  e.  ( U. k  ^m  U. j )  |  A. y  e.  k  ( `' f
" y )  e.  j } )
21a1i 11 . 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 735 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
k  =  K )
43unieqd 4028 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. k  =  U. K )
5 toponuni 16994 . . . . . 6  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
65ad2antlr 709 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  Y  =  U. K )
74, 6eqtr4d 2473 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. k  =  Y
)
8 simprl 734 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
j  =  J )
98unieqd 4028 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  U. J )
10 toponuni 16994 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1110ad2antrr 708 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  X  =  U. J )
129, 11eqtr4d 2473 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  X
)
137, 12oveq12d 6101 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
( U. k  ^m  U. j )  =  ( Y  ^m  X ) )
148eleq2d 2505 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( j  =  J  /\  k  =  K ) )  -> 
( ( `' f
" y )  e.  j  <->  ( `' f
" y )  e.  J ) )
153, 14raleqbidv 2918 . . 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 2953 . 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 16993 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1817adantr 453 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  J  e.  Top )
19 topontop 16993 . . 3  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
2019adantl 454 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  K  e.  Top )
21 ovex 6108 . . . 4  |-  ( Y  ^m  X )  e. 
_V
2221rabex 4356 . . 3  |-  { f  e.  ( Y  ^m  X )  |  A. y  e.  K  ( `' f " y
)  e.  J }  e.  _V
2322a1i 11 . 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 6203 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 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958   U.cuni 4017   `'ccnv 4879   "cima 4883   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085    ^m cmap 7020   Topctop 16960  TopOnctopon 16961    Cn ccn 17290
This theorem is referenced by:  iscn  17301  cnfex  27677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-topon 16968  df-cn 17293
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