Theorem fvmpt2curryd 7397

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.

Step	Hyp	Ref	Expression
1		fvmpt2curryd.b	. . 3 |- ( ph -> B e. Y )
2		csbcom 3994	. . . . 5 |- [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C = [_ A / a ]_ [_ B / b ]_ [_ b / y ]_ [_ a / x ]_ C
3		csbco 3543	. . . . . 6 |- [_ B / b ]_ [_ b / y ]_ [_ a / x ]_ C = [_ B / y ]_ [_ a / x ]_ C
4	3	csbeq2i 3993	. . . . 5 |- [_ A / a ]_ [_ B / b ]_ [_ b / y ]_ [_ a / x ]_ C = [_ A / a ]_ [_ B / y ]_ [_ a / x ]_ C
5		csbcom 3994	. . . . . 6 |- [_ A / a ]_ [_ B / y ]_ [_ a / x ]_ C = [_ B / y ]_ [_ A / a ]_ [_ a / x ]_ C
6		csbco 3543	. . . . . . 7 |- [_ A / a ]_ [_ a / x ]_ C = [_ A / x ]_ C
7	6	csbeq2i 3993	. . . . . 6 |- [_ B / y ]_ [_ A / a ]_ [_ a / x ]_ C = [_ B / y ]_ [_ A / x ]_ C
8	5, 7	eqtri 2644	. . . . 5 |- [_ A / a ]_ [_ B / y ]_ [_ a / x ]_ C = [_ B / y ]_ [_ A / x ]_ C
9	2, 4, 8	3eqtri 2648	. . . 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 3549	. . . . . . . 8 |- F/_ x [_ A / x ]_ C
13	12	nfel1 2779	. . . . . . 7 |- F/ x [_ A / x ]_ C e. V
14		nfcsb1v 3549	. . . . . . . 8 |- F/_ y [_ B / y ]_ [_ A / x ]_ C
15	14	nfel1 2779	. . . . . . 7 |- F/ y [_ B / y ]_ [_ A / x ]_ C e. V
16		csbeq1a 3542	. . . . . . . 8 |- ( x = A -> C = [_ A / x ]_ C )
17	16	eleq1d 2686	. . . . . . 7 |- ( x = A -> ( C e. V <-> [_ A / x ]_ C e. V ) )
18		csbeq1a 3542	. . . . . . . 8 |- ( y = B -> [_ A / x ]_ C = [_ B / y ]_ [_ A / x ]_ C )
19	18	eleq1d 2686	. . . . . . 7 |- ( y = B -> ( [_ A / x ]_ C e. V <-> [_ B / y ]_ [_ A / x ]_ C e. V ) )
20	13, 15, 17, 19	rspc2 3320	. . . . . 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 ) )
21	20	imp 445	. . . . 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 )
22	10, 1, 11, 21	syl21anc 1325	. . . 4 |- ( ph -> [_ B / y ]_ [_ A / x ]_ C e. V )
23	9, 22	syl5eqel 2705	. . 3 |- ( ph -> [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C e. V )
24		eqid 2622	. . . 4 |- ( b e. Y |-> [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C ) = ( b e. Y |-> [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C )
25	24	fvmpts 6285	. . 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 )
26	1, 23, 25	syl2anc 693	. 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 2764	. . . . . 6 |- F/_ a C
29		nfcv 2764	. . . . . 6 |- F/_ b C
30		nfcv 2764	. . . . . . 7 |- F/_ x b
31		nfcsb1v 3549	. . . . . . 7 |- F/_ x [_ a / x ]_ C
32	30, 31	nfcsb 3551	. . . . . 6 |- F/_ x [_ b / y ]_ [_ a / x ]_ C
33		nfcsb1v 3549	. . . . . 6 |- F/_ y [_ b / y ]_ [_ a / x ]_ C
34		csbeq1a 3542	. . . . . . 7 |- ( x = a -> C = [_ a / x ]_ C )
35		csbeq1a 3542	. . . . . . 7 |- ( y = b -> [_ a / x ]_ C = [_ b / y ]_ [_ a / x ]_ C )
36	34, 35	sylan9eq 2676	. . . . . 6 |- ( ( x = a /\ y = b ) -> C = [_ b / y ]_ [_ a / x ]_ C )
37	28, 29, 32, 33, 36	cbvmpt2 6734	. . . . 5 |- ( x e. X , y e. Y |-> C ) = ( a e. X , b e. Y |-> [_ b / y ]_ [_ a / x ]_ C )
38	27, 37	eqtri 2644	. . . 4 |- F = ( a e. X , b e. Y |-> [_ b / y ]_ [_ a / x ]_ C )
39	31	nfel1 2779	. . . . . . 7 |- F/ x [_ a / x ]_ C e. V
40	33	nfel1 2779	. . . . . . 7 |- F/ y [_ b / y ]_ [_ a / x ]_ C e. V
41	34	eleq1d 2686	. . . . . . 7 |- ( x = a -> ( C e. V <-> [_ a / x ]_ C e. V ) )
42	35	eleq1d 2686	. . . . . . 7 |- ( y = b -> ( [_ a / x ]_ C e. V <-> [_ b / y ]_ [_ a / x ]_ C e. V ) )
43	39, 40, 41, 42	rspc2 3320	. . . . . 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 ) )
44	11, 43	mpan9 486	. . . . 5 |- ( ( ph /\ ( a e. X /\ b e. Y ) ) -> [_ b / y ]_ [_ a / x ]_ C e. V )
45	44	ralrimivva 2971	. . . 4 |- ( ph -> A. a e. X A. b e. Y [_ b / y ]_ [_ a / x ]_ C e. V )
46		ne0i 3921	. . . . 5 |- ( B e. Y -> Y =/= (/) )
47	1, 46	syl 17	. . . 4 |- ( ph -> Y =/= (/) )
48		fvmpt2curryd.y	. . . 4 |- ( ph -> Y e. W )
49	38, 45, 47, 48, 10	mpt2curryvald 7396	. . 3 |- ( ph -> (curry F ` A ) = ( b e. Y |-> [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C ) )
50	49	fveq1d 6193	. 2 |- ( ph -> ( (curry F ` A ) ` B ) = ( ( b e. Y |-> [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C ) ` B ) )
51	27	a1i 11	. . 3 |- ( ph -> F = ( x e. X , y e. Y |-> C ) )
52		csbco 3543	. . . . . . . 8 |- [_ y / b ]_ [_ b / y ]_ [_ a / x ]_ C = [_ y / y ]_ [_ a / x ]_ C
53		csbid 3541	. . . . . . . 8 |- [_ y / y ]_ [_ a / x ]_ C = [_ a / x ]_ C
54	52, 53	eqtr2i 2645	. . . . . . 7 |- [_ a / x ]_ C = [_ y / b ]_ [_ b / y ]_ [_ a / x ]_ C
55	54	a1i 11	. . . . . 6 |- ( ph -> [_ a / x ]_ C = [_ y / b ]_ [_ b / y ]_ [_ a / x ]_ C )
56	55	csbeq2dv 3992	. . . . 5 |- ( ph -> [_ x / a ]_ [_ a / x ]_ C = [_ x / a ]_ [_ y / b ]_ [_ b / y ]_ [_ a / x ]_ C )
57		csbco 3543	. . . . . 6 |- [_ x / a ]_ [_ a / x ]_ C = [_ x / x ]_ C
58		csbid 3541	. . . . . 6 |- [_ x / x ]_ C = C
59	57, 58	eqtri 2644	. . . . 5 |- [_ x / a ]_ [_ a / x ]_ C = C
60		csbcom 3994	. . . . 5 |- [_ x / a ]_ [_ y / b ]_ [_ b / y ]_ [_ a / x ]_ C = [_ y / b ]_ [_ x / a ]_ [_ b / y ]_ [_ a / x ]_ C
61	56, 59, 60	3eqtr3g 2679	. . . 4 |- ( ph -> C = [_ y / b ]_ [_ x / a ]_ [_ b / y ]_ [_ a / x ]_ C )
62		csbeq1 3536	. . . . . . 7 |- ( x = A -> [_ x / a ]_ [_ b / y ]_ [_ a / x ]_ C = [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C )
63	62	adantr 481	. . . . . 6 |- ( ( x = A /\ y = B ) -> [_ x / a ]_ [_ b / y ]_ [_ a / x ]_ C = [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C )
64	63	csbeq2dv 3992	. . . . 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 3536	. . . . . 6 |- ( y = B -> [_ y / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C = [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C )
66	65	adantl 482	. . . . 5 |- ( ( x = A /\ y = B ) -> [_ y / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C = [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C )
67	64, 66	eqtrd 2656	. . . 4 |- ( ( x = A /\ y = B ) -> [_ y / b ]_ [_ x / a ]_ [_ b / y ]_ [_ a / x ]_ C = [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C )
68	61, 67	sylan9eq 2676	. . 3 |- ( ( ph /\ ( x = A /\ y = B ) ) -> C = [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C )
69		eqidd 2623	. . 3 |- ( ( ph /\ x = A ) -> Y = Y )
70		nfv 1843	. . 3 |- F/ x ph
71		nfv 1843	. . 3 |- F/ y ph
72		nfcv 2764	. . 3 |- F/_ y A
73		nfcv 2764	. . 3 |- F/_ x B
74		nfcv 2764	. . . . 5 |- F/_ x A
75	74, 32	nfcsb 3551	. . . 4 |- F/_ x [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C
76	73, 75	nfcsb 3551	. . 3 |- F/_ x [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C
77	9, 14	nfcxfr 2762	. . 3 |- F/_ y [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C
78	51, 68, 69, 10, 1, 23, 70, 71, 72, 73, 76, 77	ovmpt2dxf 6786	. 2 |- ( ph -> ( A F B ) = [_ B / b ]_ [_ A / a ]_ [_ b / y ]_ [_ a / x ]_ C )
79	26, 50, 78	3eqtr4d 2666	1 |- ( ph -> ( (curry F ` A ) ` B ) = ( A F B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   (/)c0 3915   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> 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-fal 1489  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:  pmatcollpw3lem  20588  logbfval  24528