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.

Step	Hyp	Ref	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 =/= (/) )
4	1, 2, 3	mpt2curryd 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
8	6, 7	nfmpt 4483	. . . 4 |- F/_ x ( y e. Y |-> [_ a / x ]_ C )
9		csbeq1a 3399	. . . . 5 |- ( x = a -> C = [_ a / x ]_ C )
10	9	mpteq2dv 4482	. . . 4 |- ( x = a -> ( y e. Y |-> C ) = ( y e. Y |-> [_ a / x ]_ C ) )
11	5, 8, 10	cbvmpt 4485	. . 3 |- ( x e. X |-> ( y e. Y |-> C ) ) = ( a e. X |-> ( y e. Y |-> [_ a / x ]_ C ) )
12	4, 11	syl6eq 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 )
14	13	adantl 466	. . 3 |- ( ( ph /\ a = A ) -> [_ a / x ]_ C = [_ A / x ]_ C )
15	14	mpteq2dv 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 )
19	17, 18	syl 16	. 2 |- ( ph -> ( y e. Y |-> [_ A / x ]_ C ) e. _V )
20	12, 15, 16, 19	fvmptd 5883	1 |- ( ph -> (curry F ` A ) = ( y e. Y |-> [_ A / x ]_ C ) )

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