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Theorem mpt2curryvald 6991
Description: The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.)
Hypotheses
Ref Expression
mpt2curryd.f  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )
mpt2curryd.c  |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  C  e.  V )
mpt2curryd.n  |-  ( ph  ->  Y  =/=  (/) )
mpt2curryvald.y  |-  ( ph  ->  Y  e.  W )
mpt2curryvald.a  |-  ( ph  ->  A  e.  X )
Assertion
Ref Expression
mpt2curryvald  |-  ( ph  ->  (curry  F `  A
)  =  ( y  e.  Y  |->  [_ A  /  x ]_ C ) )
Distinct variable groups:    x, F, y    x, V, y    x, X, y    x, Y, y    ph, x, y    x, A, y
Allowed substitution hints:    C( x, y)    W( x, y)

Proof of Theorem mpt2curryvald
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 mpt2curryd.f . . . 4  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )
2 mpt2curryd.c . . . 4  |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  C  e.  V )
3 mpt2curryd.n . . . 4  |-  ( ph  ->  Y  =/=  (/) )
41, 2, 3mpt2curryd 6990 . . 3  |-  ( ph  -> curry 
F  =  ( x  e.  X  |->  ( y  e.  Y  |->  C ) ) )
5 nfcv 2616 . . . 4  |-  F/_ a
( y  e.  Y  |->  C )
6 nfcv 2616 . . . . 5  |-  F/_ x Y
7 nfcsb1v 3436 . . . . 5  |-  F/_ x [_ a  /  x ]_ C
86, 7nfmpt 4527 . . . 4  |-  F/_ x
( y  e.  Y  |-> 
[_ a  /  x ]_ C )
9 csbeq1a 3429 . . . . 5  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
109mpteq2dv 4526 . . . 4  |-  ( x  =  a  ->  (
y  e.  Y  |->  C )  =  ( y  e.  Y  |->  [_ a  /  x ]_ C ) )
115, 8, 10cbvmpt 4529 . . 3  |-  ( x  e.  X  |->  ( y  e.  Y  |->  C ) )  =  ( a  e.  X  |->  ( y  e.  Y  |->  [_ a  /  x ]_ C ) )
124, 11syl6eq 2511 . 2  |-  ( ph  -> curry 
F  =  ( a  e.  X  |->  ( y  e.  Y  |->  [_ a  /  x ]_ C ) ) )
13 csbeq1 3423 . . . 4  |-  ( a  =  A  ->  [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C )
1413adantl 464 . . 3  |-  ( (
ph  /\  a  =  A )  ->  [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C )
1514mpteq2dv 4526 . 2  |-  ( (
ph  /\  a  =  A )  ->  (
y  e.  Y  |->  [_ a  /  x ]_ C
)  =  ( y  e.  Y  |->  [_ A  /  x ]_ C ) )
16 mpt2curryvald.a . 2  |-  ( ph  ->  A  e.  X )
17 mpt2curryvald.y . . 3  |-  ( ph  ->  Y  e.  W )
18 mptexg 6117 . . 3  |-  ( Y  e.  W  ->  (
y  e.  Y  |->  [_ A  /  x ]_ C
)  e.  _V )
1917, 18syl 16 . 2  |-  ( ph  ->  ( y  e.  Y  |-> 
[_ A  /  x ]_ C )  e.  _V )
2012, 15, 16, 19fvmptd 5936 1  |-  ( ph  ->  (curry  F `  A
)  =  ( y  e.  Y  |->  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106   [_csb 3420   (/)c0 3783    |-> cmpt 4497   ` cfv 5570    |-> cmpt2 6272  curry ccur 6986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-cur 6988
This theorem is referenced by:  fvmpt2curryd  6992  pmatcollpw3lem  19451  logbmpt  23327
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