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Theorem mpt2curryvald 6894
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 6893 . . 3  |-  ( ph  -> curry 
F  =  ( x  e.  X  |->  ( y  e.  Y  |->  C ) ) )
5 nfcv 2614 . . . 4  |-  F/_ a
( y  e.  Y  |->  C )
6 nfcv 2614 . . . . 5  |-  F/_ x Y
7 nfcsb1v 3406 . . . . 5  |-  F/_ x [_ a  /  x ]_ C
86, 7nfmpt 4483 . . . 4  |-  F/_ x
( y  e.  Y  |-> 
[_ a  /  x ]_ C )
9 csbeq1a 3399 . . . . 5  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
109mpteq2dv 4482 . . . 4  |-  ( x  =  a  ->  (
y  e.  Y  |->  C )  =  ( y  e.  Y  |->  [_ a  /  x ]_ C ) )
115, 8, 10cbvmpt 4485 . . 3  |-  ( x  e.  X  |->  ( y  e.  Y  |->  C ) )  =  ( a  e.  X  |->  ( y  e.  Y  |->  [_ a  /  x ]_ C ) )
124, 11syl6eq 2509 . 2  |-  ( ph  -> curry 
F  =  ( a  e.  X  |->  ( y  e.  Y  |->  [_ a  /  x ]_ C ) ) )
13 csbeq1 3393 . . . 4  |-  ( a  =  A  ->  [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C )
1413adantl 466 . . 3  |-  ( (
ph  /\  a  =  A )  ->  [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C )
1514mpteq2dv 4482 . 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 6051 . . 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 5883 1  |-  ( ph  ->  (curry  F `  A
)  =  ( y  e.  Y  |->  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   _Vcvv 3072   [_csb 3390   (/)c0 3740    |-> cmpt 4453   ` cfv 5521    |-> cmpt2 6197  curry ccur 6889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-cur 6891
This theorem is referenced by:  fvmpt2curryd  6895  pmatcollpw3  31252  pmatcollpw3fi  31253
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