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Theorem fvmpt2curryd 7018
Description: The value of the value of a curried operation given in maps-to notation is the operation value of the original operation. (Contributed by AV, 27-Oct-2019.)
Hypotheses
Ref Expression
fvmpt2curryd.f  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )
fvmpt2curryd.c  |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  C  e.  V )
fvmpt2curryd.y  |-  ( ph  ->  Y  e.  W )
fvmpt2curryd.a  |-  ( ph  ->  A  e.  X )
fvmpt2curryd.b  |-  ( ph  ->  B  e.  Y )
Assertion
Ref Expression
fvmpt2curryd  |-  ( ph  ->  ( (curry  F `  A ) `  B
)  =  ( A F B ) )
Distinct variable groups:    x, A, y    x, B, y    x, V, y    x, X, y   
x, Y, y    ph, x, y
Allowed substitution hints:    C( x, y)    F( x, y)    W( x, y)

Proof of Theorem fvmpt2curryd
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvmpt2curryd.b . . 3  |-  ( ph  ->  B  e.  Y )
2 csbcom 3845 . . . . 5  |-  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  = 
[_ A  /  a ]_ [_ B  /  b ]_ [_ b  /  y ]_ [_ a  /  x ]_ C
3 csbco 3440 . . . . . 6  |-  [_ B  /  b ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  = 
[_ B  /  y ]_ [_ a  /  x ]_ C
43csbeq2i 3844 . . . . 5  |-  [_ A  /  a ]_ [_ B  /  b ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  = 
[_ A  /  a ]_ [_ B  /  y ]_ [_ a  /  x ]_ C
5 csbcom 3845 . . . . . 6  |-  [_ A  /  a ]_ [_ B  /  y ]_ [_ a  /  x ]_ C  = 
[_ B  /  y ]_ [_ A  /  a ]_ [_ a  /  x ]_ C
6 csbco 3440 . . . . . . 7  |-  [_ A  /  a ]_ [_ a  /  x ]_ C  = 
[_ A  /  x ]_ C
76csbeq2i 3844 . . . . . 6  |-  [_ B  /  y ]_ [_ A  /  a ]_ [_ a  /  x ]_ C  = 
[_ B  /  y ]_ [_ A  /  x ]_ C
85, 7eqtri 2486 . . . . 5  |-  [_ A  /  a ]_ [_ B  /  y ]_ [_ a  /  x ]_ C  = 
[_ B  /  y ]_ [_ A  /  x ]_ C
92, 4, 83eqtri 2490 . . . 4  |-  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  = 
[_ B  /  y ]_ [_ A  /  x ]_ C
10 fvmpt2curryd.a . . . . 5  |-  ( ph  ->  A  e.  X )
11 fvmpt2curryd.c . . . . 5  |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  C  e.  V )
12 nfcsb1v 3446 . . . . . . . 8  |-  F/_ x [_ A  /  x ]_ C
1312nfel1 2635 . . . . . . 7  |-  F/ x [_ A  /  x ]_ C  e.  V
14 nfcsb1v 3446 . . . . . . . 8  |-  F/_ y [_ B  /  y ]_ [_ A  /  x ]_ C
1514nfel1 2635 . . . . . . 7  |-  F/ y
[_ B  /  y ]_ [_ A  /  x ]_ C  e.  V
16 csbeq1a 3439 . . . . . . . 8  |-  ( x  =  A  ->  C  =  [_ A  /  x ]_ C )
1716eleq1d 2526 . . . . . . 7  |-  ( x  =  A  ->  ( C  e.  V  <->  [_ A  /  x ]_ C  e.  V
) )
18 csbeq1a 3439 . . . . . . . 8  |-  ( y  =  B  ->  [_ A  /  x ]_ C  = 
[_ B  /  y ]_ [_ A  /  x ]_ C )
1918eleq1d 2526 . . . . . . 7  |-  ( y  =  B  ->  ( [_ A  /  x ]_ C  e.  V  <->  [_ B  /  y ]_ [_ A  /  x ]_ C  e.  V )
)
2013, 15, 17, 19rspc2 3218 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( A. x  e.  X  A. y  e.  Y  C  e.  V  ->  [_ B  /  y ]_ [_ A  /  x ]_ C  e.  V
) )
2120imp 429 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A. x  e.  X  A. y  e.  Y  C  e.  V )  ->  [_ B  /  y ]_ [_ A  /  x ]_ C  e.  V )
2210, 1, 11, 21syl21anc 1227 . . . 4  |-  ( ph  ->  [_ B  /  y ]_ [_ A  /  x ]_ C  e.  V
)
239, 22syl5eqel 2549 . . 3  |-  ( ph  ->  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  e.  V
)
24 eqid 2457 . . . 4  |-  ( b  e.  Y  |->  [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )  =  ( b  e.  Y  |->  [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
2524fvmpts 5958 . . 3  |-  ( ( B  e.  Y  /\  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  e.  V )  ->  ( ( b  e.  Y  |->  [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C ) `  B
)  =  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
261, 23, 25syl2anc 661 . 2  |-  ( ph  ->  ( ( b  e.  Y  |->  [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C ) `  B
)  =  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
27 fvmpt2curryd.f . . . . 5  |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )
28 nfcv 2619 . . . . . 6  |-  F/_ a C
29 nfcv 2619 . . . . . 6  |-  F/_ b C
30 nfcv 2619 . . . . . . 7  |-  F/_ x
b
31 nfcsb1v 3446 . . . . . . 7  |-  F/_ x [_ a  /  x ]_ C
3230, 31nfcsb 3448 . . . . . 6  |-  F/_ x [_ b  /  y ]_ [_ a  /  x ]_ C
33 nfcsb1v 3446 . . . . . 6  |-  F/_ y [_ b  /  y ]_ [_ a  /  x ]_ C
34 csbeq1a 3439 . . . . . . 7  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
35 csbeq1a 3439 . . . . . . 7  |-  ( y  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  y ]_ [_ a  /  x ]_ C )
3634, 35sylan9eq 2518 . . . . . 6  |-  ( ( x  =  a  /\  y  =  b )  ->  C  =  [_ b  /  y ]_ [_ a  /  x ]_ C )
3728, 29, 32, 33, 36cbvmpt2 6375 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  C )  =  ( a  e.  X ,  b  e.  Y  |->  [_ b  /  y ]_ [_ a  /  x ]_ C )
3827, 37eqtri 2486 . . . 4  |-  F  =  ( a  e.  X ,  b  e.  Y  |-> 
[_ b  /  y ]_ [_ a  /  x ]_ C )
3931nfel1 2635 . . . . . . 7  |-  F/ x [_ a  /  x ]_ C  e.  V
4033nfel1 2635 . . . . . . 7  |-  F/ y
[_ b  /  y ]_ [_ a  /  x ]_ C  e.  V
4134eleq1d 2526 . . . . . . 7  |-  ( x  =  a  ->  ( C  e.  V  <->  [_ a  /  x ]_ C  e.  V
) )
4235eleq1d 2526 . . . . . . 7  |-  ( y  =  b  ->  ( [_ a  /  x ]_ C  e.  V  <->  [_ b  /  y ]_ [_ a  /  x ]_ C  e.  V )
)
4339, 40, 41, 42rspc2 3218 . . . . . 6  |-  ( ( a  e.  X  /\  b  e.  Y )  ->  ( A. x  e.  X  A. y  e.  Y  C  e.  V  ->  [_ b  /  y ]_ [_ a  /  x ]_ C  e.  V
) )
4411, 43mpan9 469 . . . . 5  |-  ( (
ph  /\  ( a  e.  X  /\  b  e.  Y ) )  ->  [_ b  /  y ]_ [_ a  /  x ]_ C  e.  V
)
4544ralrimivva 2878 . . . 4  |-  ( ph  ->  A. a  e.  X  A. b  e.  Y  [_ b  /  y ]_ [_ a  /  x ]_ C  e.  V )
46 ne0i 3799 . . . . 5  |-  ( B  e.  Y  ->  Y  =/=  (/) )
471, 46syl 16 . . . 4  |-  ( ph  ->  Y  =/=  (/) )
48 fvmpt2curryd.y . . . 4  |-  ( ph  ->  Y  e.  W )
4938, 45, 47, 48, 10mpt2curryvald 7017 . . 3  |-  ( ph  ->  (curry  F `  A
)  =  ( b  e.  Y  |->  [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C ) )
5049fveq1d 5874 . 2  |-  ( ph  ->  ( (curry  F `  A ) `  B
)  =  ( ( b  e.  Y  |->  [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C ) `
 B ) )
5127a1i 11 . . 3  |-  ( ph  ->  F  =  ( x  e.  X ,  y  e.  Y  |->  C ) )
52 csbco 3440 . . . . . . . 8  |-  [_ y  /  b ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  = 
[_ y  /  y ]_ [_ a  /  x ]_ C
53 csbid 3438 . . . . . . . 8  |-  [_ y  /  y ]_ [_ a  /  x ]_ C  = 
[_ a  /  x ]_ C
5452, 53eqtr2i 2487 . . . . . . 7  |-  [_ a  /  x ]_ C  = 
[_ y  /  b ]_ [_ b  /  y ]_ [_ a  /  x ]_ C
5554a1i 11 . . . . . 6  |-  ( ph  ->  [_ a  /  x ]_ C  =  [_ y  /  b ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
5655csbeq2dv 3843 . . . . 5  |-  ( ph  ->  [_ x  /  a ]_ [_ a  /  x ]_ C  =  [_ x  /  a ]_ [_ y  /  b ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
57 csbco 3440 . . . . . 6  |-  [_ x  /  a ]_ [_ a  /  x ]_ C  = 
[_ x  /  x ]_ C
58 csbid 3438 . . . . . 6  |-  [_ x  /  x ]_ C  =  C
5957, 58eqtri 2486 . . . . 5  |-  [_ x  /  a ]_ [_ a  /  x ]_ C  =  C
60 csbcom 3845 . . . . 5  |-  [_ x  /  a ]_ [_ y  /  b ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  = 
[_ y  /  b ]_ [_ x  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C
6156, 59, 603eqtr3g 2521 . . . 4  |-  ( ph  ->  C  =  [_ y  /  b ]_ [_ x  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
62 csbeq1 3433 . . . . . . 7  |-  ( x  =  A  ->  [_ x  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  = 
[_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
6362adantr 465 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  [_ x  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  =  [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
6463csbeq2dv 3843 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  [_ y  /  b ]_ [_ x  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  =  [_ y  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
65 csbeq1 3433 . . . . . 6  |-  ( y  =  B  ->  [_ y  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  = 
[_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
6665adantl 466 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  [_ y  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  =  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
6764, 66eqtrd 2498 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  [_ y  /  b ]_ [_ x  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C  =  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
6861, 67sylan9eq 2518 . . 3  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  C  =  [_ B  / 
b ]_ [_ A  / 
a ]_ [_ b  / 
y ]_ [_ a  /  x ]_ C )
69 eqidd 2458 . . 3  |-  ( (
ph  /\  x  =  A )  ->  Y  =  Y )
70 nfv 1708 . . 3  |-  F/ x ph
71 nfv 1708 . . 3  |-  F/ y
ph
72 nfcv 2619 . . 3  |-  F/_ y A
73 nfcv 2619 . . 3  |-  F/_ x B
74 nfcv 2619 . . . . 5  |-  F/_ x A
7574, 32nfcsb 3448 . . . 4  |-  F/_ x [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C
7673, 75nfcsb 3448 . . 3  |-  F/_ x [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C
779, 14nfcxfr 2617 . . 3  |-  F/_ y [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C
7851, 68, 69, 10, 1, 23, 70, 71, 72, 73, 76, 77ovmpt2dxf 6427 . 2  |-  ( ph  ->  ( A F B )  =  [_ B  /  b ]_ [_ A  /  a ]_ [_ b  /  y ]_ [_ a  /  x ]_ C )
7926, 50, 783eqtr4d 2508 1  |-  ( ph  ->  ( (curry  F `  A ) `  B
)  =  ( A F B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   [_csb 3430   (/)c0 3793    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298  curry ccur 7012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-cur 7014
This theorem is referenced by:  pmatcollpw3lem  19411
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