Theorem mpt2curryvald 7396

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 7395	. . 3 |- ( ph -> curry F = ( x e. X |-> ( y e. Y |-> C ) ) )
5		nfcv 2764	. . . 4 |- F/_ a ( y e. Y |-> C )
6		nfcv 2764	. . . . 5 |- F/_ x Y
7		nfcsb1v 3549	. . . . 5 |- F/_ x [_ a / x ]_ C
8	6, 7	nfmpt 4746	. . . 4 |- F/_ x ( y e. Y |-> [_ a / x ]_ C )
9		csbeq1a 3542	. . . . 5 |- ( x = a -> C = [_ a / x ]_ C )
10	9	mpteq2dv 4745	. . . 4 |- ( x = a -> ( y e. Y |-> C ) = ( y e. Y |-> [_ a / x ]_ C ) )
11	5, 8, 10	cbvmpt 4749	. . 3 |- ( x e. X |-> ( y e. Y |-> C ) ) = ( a e. X |-> ( y e. Y |-> [_ a / x ]_ C ) )
12	4, 11	syl6eq 2672	. 2 |- ( ph -> curry F = ( a e. X |-> ( y e. Y |-> [_ a / x ]_ C ) ) )
13		csbeq1 3536	. . . 4 |- ( a = A -> [_ a / x ]_ C = [_ A / x ]_ C )
14	13	adantl 482	. . 3 |- ( ( ph /\ a = A ) -> [_ a / x ]_ C = [_ A / x ]_ C )
15	14	mpteq2dv 4745	. 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 6484	. . 3 |- ( Y e. W -> ( y e. Y |-> [_ A / x ]_ C ) e. _V )
19	17, 18	syl 17	. 2 |- ( ph -> ( y e. Y |-> [_ A / x ]_ C ) e. _V )
20	12, 15, 16, 19	fvmptd 6288	1 |- ( ph -> (curry F ` A ) = ( y e. Y |-> [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   [_csb 3533   (/)c0 3915   |-> cmpt 4729   ` cfv 5888   |-> cmpt2 6652  curry ccur 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cur 7393

This theorem is referenced by:  fvmpt2curryd  7397  pmatcollpw3lem  20588  logbmpt  24526