Theorem offval22 7253

Description: The function operation expressed as a mapping, variation of offval2 6914. (Contributed by SO, 15-Jul-2018.)

Hypotheses

Ref	Expression
offval22.a	|- ( ph -> A e. V )
offval22.b	|- ( ph -> B e. W )
offval22.c	|- ( ( ph /\ x e. A /\ y e. B ) -> C e. X )
offval22.d	|- ( ( ph /\ x e. A /\ y e. B ) -> D e. Y )
offval22.f	|- ( ph -> F = ( x e. A , y e. B |-> C ) )
offval22.g	|- ( ph -> G = ( x e. A , y e. B |-> D ) )

Assertion

Ref	Expression
offval22	|- ( ph -> ( F oF R G ) = ( x e. A , y e. B |-> ( C R D ) ) )

Distinct variable groups:   ph, x, y   x, A, y   x, B, y   x, R, y

Allowed substitution hints:   C( x, y)   D( x, y)   F( x, y)   G( x, y)   V( x, y)   W( x, y)   X( x, y)   Y( x, y)

Proof of Theorem offval22

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval22.a	. . . 4 |- ( ph -> A e. V )
2		offval22.b	. . . 4 |- ( ph -> B e. W )
3		xpexg 6960	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ph -> ( A X. B ) e. _V )
5		xp1st 7198	. . . . 5 |- ( z e. ( A X. B ) -> ( 1st ` z ) e. A )
6		xp2nd 7199	. . . . 5 |- ( z e. ( A X. B ) -> ( 2nd ` z ) e. B )
7	5, 6	jca 554	. . . 4 |- ( z e. ( A X. B ) -> ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) )
8		fvex 6201	. . . . . 6 |- ( 2nd ` z ) e. _V
9		fvex 6201	. . . . . 6 |- ( 1st ` z ) e. _V
10		nfcv 2764	. . . . . . 7 |- F/_ y ( 2nd ` z )
11		nfcv 2764	. . . . . . 7 |- F/_ x ( 2nd ` z )
12		nfcv 2764	. . . . . . 7 |- F/_ x ( 1st ` z )
13		nfv 1843	. . . . . . . 8 |- F/ y ( ph /\ x e. A /\ ( 2nd ` z ) e. B )
14		nfcsb1v 3549	. . . . . . . . 9 |- F/_ y [_ ( 2nd ` z ) / y ]_ C
15	14	nfel1 2779	. . . . . . . 8 |- F/ y [_ ( 2nd ` z ) / y ]_ C e. _V
16	13, 15	nfim 1825	. . . . . . 7 |- F/ y ( ( ph /\ x e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 2nd ` z ) / y ]_ C e. _V )
17		nfv 1843	. . . . . . . 8 |- F/ x ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B )
18		nfcsb1v 3549	. . . . . . . . 9 |- F/_ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C
19	18	nfel1 2779	. . . . . . . 8 |- F/ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V
20	17, 19	nfim 1825	. . . . . . 7 |- F/ x ( ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
21		eleq1 2689	. . . . . . . . 9 |- ( y = ( 2nd ` z ) -> ( y e. B <-> ( 2nd ` z ) e. B ) )
22	21	3anbi3d 1405	. . . . . . . 8 |- ( y = ( 2nd ` z ) -> ( ( ph /\ x e. A /\ y e. B ) <-> ( ph /\ x e. A /\ ( 2nd ` z ) e. B ) ) )
23		csbeq1a 3542	. . . . . . . . 9 |- ( y = ( 2nd ` z ) -> C = [_ ( 2nd ` z ) / y ]_ C )
24	23	eleq1d 2686	. . . . . . . 8 |- ( y = ( 2nd ` z ) -> ( C e. _V <-> [_ ( 2nd ` z ) / y ]_ C e. _V ) )
25	22, 24	imbi12d 334	. . . . . . 7 |- ( y = ( 2nd ` z ) -> ( ( ( ph /\ x e. A /\ y e. B ) -> C e. _V ) <-> ( ( ph /\ x e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 2nd ` z ) / y ]_ C e. _V ) ) )
26		eleq1 2689	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( x e. A <-> ( 1st ` z ) e. A ) )
27	26	3anbi2d 1404	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( ( ph /\ x e. A /\ ( 2nd ` z ) e. B ) <-> ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) ) )
28		csbeq1a 3542	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> [_ ( 2nd ` z ) / y ]_ C = [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
29	28	eleq1d 2686	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( [_ ( 2nd ` z ) / y ]_ C e. _V <-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) )
30	27, 29	imbi12d 334	. . . . . . 7 |- ( x = ( 1st ` z ) -> ( ( ( ph /\ x e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 2nd ` z ) / y ]_ C e. _V ) <-> ( ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) ) )
31		offval22.c	. . . . . . . 8 |- ( ( ph /\ x e. A /\ y e. B ) -> C e. X )
32		elex 3212	. . . . . . . 8 |- ( C e. X -> C e. _V )
33	31, 32	syl 17	. . . . . . 7 |- ( ( ph /\ x e. A /\ y e. B ) -> C e. _V )
34	10, 11, 12, 16, 20, 25, 30, 33	vtocl2gf 3268	. . . . . 6 |- ( ( ( 2nd ` z ) e. _V /\ ( 1st ` z ) e. _V ) -> ( ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V ) )
35	8, 9, 34	mp2an 708	. . . . 5 |- ( ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
36	35	3expb 1266	. . . 4 |- ( ( ph /\ ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
37	7, 36	sylan2 491	. . 3 |- ( ( ph /\ z e. ( A X. B ) ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C e. _V )
38		nfcsb1v 3549	. . . . . . . . 9 |- F/_ y [_ ( 2nd ` z ) / y ]_ D
39	38	nfel1 2779	. . . . . . . 8 |- F/ y [_ ( 2nd ` z ) / y ]_ D e. _V
40	13, 39	nfim 1825	. . . . . . 7 |- F/ y ( ( ph /\ x e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 2nd ` z ) / y ]_ D e. _V )
41		nfcsb1v 3549	. . . . . . . . 9 |- F/_ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D
42	41	nfel1 2779	. . . . . . . 8 |- F/ x [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D e. _V
43	17, 42	nfim 1825	. . . . . . 7 |- F/ x ( ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D e. _V )
44		csbeq1a 3542	. . . . . . . . 9 |- ( y = ( 2nd ` z ) -> D = [_ ( 2nd ` z ) / y ]_ D )
45	44	eleq1d 2686	. . . . . . . 8 |- ( y = ( 2nd ` z ) -> ( D e. _V <-> [_ ( 2nd ` z ) / y ]_ D e. _V ) )
46	22, 45	imbi12d 334	. . . . . . 7 |- ( y = ( 2nd ` z ) -> ( ( ( ph /\ x e. A /\ y e. B ) -> D e. _V ) <-> ( ( ph /\ x e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 2nd ` z ) / y ]_ D e. _V ) ) )
47		csbeq1a 3542	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> [_ ( 2nd ` z ) / y ]_ D = [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D )
48	47	eleq1d 2686	. . . . . . . 8 |- ( x = ( 1st ` z ) -> ( [_ ( 2nd ` z ) / y ]_ D e. _V <-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D e. _V ) )
49	27, 48	imbi12d 334	. . . . . . 7 |- ( x = ( 1st ` z ) -> ( ( ( ph /\ x e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 2nd ` z ) / y ]_ D e. _V ) <-> ( ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D e. _V ) ) )
50		offval22.d	. . . . . . . 8 |- ( ( ph /\ x e. A /\ y e. B ) -> D e. Y )
51		elex 3212	. . . . . . . 8 |- ( D e. Y -> D e. _V )
52	50, 51	syl 17	. . . . . . 7 |- ( ( ph /\ x e. A /\ y e. B ) -> D e. _V )
53	10, 11, 12, 40, 43, 46, 49, 52	vtocl2gf 3268	. . . . . 6 |- ( ( ( 2nd ` z ) e. _V /\ ( 1st ` z ) e. _V ) -> ( ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D e. _V ) )
54	8, 9, 53	mp2an 708	. . . . 5 |- ( ( ph /\ ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D e. _V )
55	54	3expb 1266	. . . 4 |- ( ( ph /\ ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D e. _V )
56	7, 55	sylan2 491	. . 3 |- ( ( ph /\ z e. ( A X. B ) ) -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D e. _V )
57		offval22.f	. . . 4 |- ( ph -> F = ( x e. A , y e. B |-> C ) )
58		mpt2mpts 7234	. . . 4 |- ( x e. A , y e. B |-> C ) = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
59	57, 58	syl6eq 2672	. . 3 |- ( ph -> F = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) )
60		offval22.g	. . . 4 |- ( ph -> G = ( x e. A , y e. B |-> D ) )
61		mpt2mpts 7234	. . . 4 |- ( x e. A , y e. B |-> D ) = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D )
62	60, 61	syl6eq 2672	. . 3 |- ( ph -> G = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D ) )
63	4, 37, 56, 59, 62	offval2 6914	. 2 |- ( ph -> ( F oF R G ) = ( z e. ( A X. B ) |-> ( [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C R [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D ) ) )
64		csbov12g 6689	. . . . . . 7 |- ( ( 2nd ` z ) e. _V -> [_ ( 2nd ` z ) / y ]_ ( C R D ) = ( [_ ( 2nd ` z ) / y ]_ C R [_ ( 2nd ` z ) / y ]_ D ) )
65	64	csbeq2dv 3992	. . . . . 6 |- ( ( 2nd ` z ) e. _V -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ ( C R D ) = [_ ( 1st ` z ) / x ]_ ( [_ ( 2nd ` z ) / y ]_ C R [_ ( 2nd ` z ) / y ]_ D ) )
66	8, 65	ax-mp 5	. . . . 5 |- [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ ( C R D ) = [_ ( 1st ` z ) / x ]_ ( [_ ( 2nd ` z ) / y ]_ C R [_ ( 2nd ` z ) / y ]_ D )
67		csbov12g 6689	. . . . . 6 |- ( ( 1st ` z ) e. _V -> [_ ( 1st ` z ) / x ]_ ( [_ ( 2nd ` z ) / y ]_ C R [_ ( 2nd ` z ) / y ]_ D ) = ( [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C R [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D ) )
68	9, 67	ax-mp 5	. . . . 5 |- [_ ( 1st ` z ) / x ]_ ( [_ ( 2nd ` z ) / y ]_ C R [_ ( 2nd ` z ) / y ]_ D ) = ( [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C R [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D )
69	66, 68	eqtr2i 2645	. . . 4 |- ( [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C R [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D ) = [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ ( C R D )
70	69	mpteq2i 4741	. . 3 |- ( z e. ( A X. B ) |-> ( [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C R [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D ) ) = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ ( C R D ) )
71		mpt2mpts 7234	. . 3 |- ( x e. A , y e. B |-> ( C R D ) ) = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ ( C R D ) )
72	70, 71	eqtr4i 2647	. 2 |- ( z e. ( A X. B ) |-> ( [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C R [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ D ) ) = ( x e. A , y e. B |-> ( C R D ) )
73	63, 72	syl6eq 2672	1 |- ( ph -> ( F oF R G ) = ( x e. A , y e. B |-> ( C R D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   1stc1st 7166   2ndc2nd 7167

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-pw 4160  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-of 6897  df-1st 7168  df-2nd 7169

This theorem is referenced by:  matsc  20256  mdetrsca2  20410  mdetrlin2  20413  mdetunilem5  20422  smadiadetglem2  20478  mat2pmatghm  20535  pm2mpghm  20621