Theorem fmpt2co 5857

Description: Composition of two functions. Variation of fmptco 5351 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypotheses

Ref	Expression
fmpt2co.1	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> R e. C )
fmpt2co.2	|- ( ph -> F = ( x e. A , y e. B |-> R ) )
fmpt2co.3	|- ( ph -> G = ( z e. C |-> S ) )
fmpt2co.4	|- ( z = R -> S = T )

Assertion

Ref	Expression
fmpt2co	|- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) )

Distinct variable groups:   x, y, B   x, z, C, y   ph, x, y   x, S, y   x, A, y   z, R   z, T

Allowed substitution hints:   ph( z)   A( z)   B( z)   R( x, y)   S( z)   T( x, y)   F( x, y, z)   G( x, y, z)

Proof of Theorem fmpt2co

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmpt2co.1	. . . . . 6 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> R e. C )
2	1	ralrimivva 2443	. . . . 5 |- ( ph -> A. x e. A A. y e. B R e. C )
3		eqid 2081	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( x e. A , y e. B |-> R )
4	3	fmpt2 5847	. . . . 5 |- ( A. x e. A A. y e. B R e. C <-> ( x e. A , y e. B |-> R ) : ( A X. B ) --> C )
5	2, 4	sylib 120	. . . 4 |- ( ph -> ( x e. A , y e. B |-> R ) : ( A X. B ) --> C )
6		nfcv 2219	. . . . . . 7 |- F/_ u R
7		nfcv 2219	. . . . . . 7 |- F/_ v R
8		nfcv 2219	. . . . . . . 8 |- F/_ x v
9		nfcsb1v 2938	. . . . . . . 8 |- F/_ x [_ u / x ]_ R
10	8, 9	nfcsb 2940	. . . . . . 7 |- F/_ x [_ v / y ]_ [_ u / x ]_ R
11		nfcsb1v 2938	. . . . . . 7 |- F/_ y [_ v / y ]_ [_ u / x ]_ R
12		csbeq1a 2916	. . . . . . . 8 |- ( x = u -> R = [_ u / x ]_ R )
13		csbeq1a 2916	. . . . . . . 8 |- ( y = v -> [_ u / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
14	12, 13	sylan9eq 2133	. . . . . . 7 |- ( ( x = u /\ y = v ) -> R = [_ v / y ]_ [_ u / x ]_ R )
15	6, 7, 10, 11, 14	cbvmpt2 5603	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( u e. A , v e. B |-> [_ v / y ]_ [_ u / x ]_ R )
16		vex 2604	. . . . . . . . . 10 |- u e. _V
17		vex 2604	. . . . . . . . . 10 |- v e. _V
18	16, 17	op2ndd 5796	. . . . . . . . 9 |- ( w = <. u , v >. -> ( 2nd ` w ) = v )
19	18	csbeq1d 2914	. . . . . . . 8 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R )
20	16, 17	op1std 5795	. . . . . . . . . 10 |- ( w = <. u , v >. -> ( 1st ` w ) = u )
21	20	csbeq1d 2914	. . . . . . . . 9 |- ( w = <. u , v >. -> [_ ( 1st ` w ) / x ]_ R = [_ u / x ]_ R )
22	21	csbeq2dv 2931	. . . . . . . 8 |- ( w = <. u , v >. -> [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
23	19, 22	eqtrd 2113	. . . . . . 7 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
24	23	mpt2mpt 5616	. . . . . 6 |- ( w e. ( A X. B ) |-> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R ) = ( u e. A , v e. B |-> [_ v / y ]_ [_ u / x ]_ R )
25	15, 24	eqtr4i 2104	. . . . 5 |- ( x e. A , y e. B |-> R ) = ( w e. ( A X. B ) |-> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R )
26	25	fmpt 5340	. . . 4 |- ( A. w e. ( A X. B ) [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R e. C <-> ( x e. A , y e. B |-> R ) : ( A X. B ) --> C )
27	5, 26	sylibr 132	. . 3 |- ( ph -> A. w e. ( A X. B ) [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R e. C )
28		fmpt2co.2	. . . 4 |- ( ph -> F = ( x e. A , y e. B |-> R ) )
29	28, 25	syl6eq 2129	. . 3 |- ( ph -> F = ( w e. ( A X. B ) |-> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R ) )
30		fmpt2co.3	. . 3 |- ( ph -> G = ( z e. C |-> S ) )
31	27, 29, 30	fmptcos 5353	. 2 |- ( ph -> ( G o. F ) = ( w e. ( A X. B ) |-> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S ) )
32	23	csbeq1d 2914	. . . . 5 |- ( w = <. u , v >. -> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
33	32	mpt2mpt 5616	. . . 4 |- ( w e. ( A X. B ) |-> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S ) = ( u e. A , v e. B |-> [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
34		nfcv 2219	. . . . 5 |- F/_ u [_ R / z ]_ S
35		nfcv 2219	. . . . 5 |- F/_ v [_ R / z ]_ S
36		nfcv 2219	. . . . . 6 |- F/_ x S
37	10, 36	nfcsb 2940	. . . . 5 |- F/_ x [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
38		nfcv 2219	. . . . . 6 |- F/_ y S
39	11, 38	nfcsb 2940	. . . . 5 |- F/_ y [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
40	14	csbeq1d 2914	. . . . 5 |- ( ( x = u /\ y = v ) -> [_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
41	34, 35, 37, 39, 40	cbvmpt2 5603	. . . 4 |- ( x e. A , y e. B |-> [_ R / z ]_ S ) = ( u e. A , v e. B |-> [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
42	33, 41	eqtr4i 2104	. . 3 |- ( w e. ( A X. B ) |-> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S ) = ( x e. A , y e. B |-> [_ R / z ]_ S )
43	1	3impb 1134	. . . . 5 |- ( ( ph /\ x e. A /\ y e. B ) -> R e. C )
44		nfcvd 2220	. . . . . 6 |- ( R e. C -> F/_ z T )
45		fmpt2co.4	. . . . . 6 |- ( z = R -> S = T )
46	44, 45	csbiegf 2946	. . . . 5 |- ( R e. C -> [_ R / z ]_ S = T )
47	43, 46	syl 14	. . . 4 |- ( ( ph /\ x e. A /\ y e. B ) -> [_ R / z ]_ S = T )
48	47	mpt2eq3dva 5589	. . 3 |- ( ph -> ( x e. A , y e. B |-> [_ R / z ]_ S ) = ( x e. A , y e. B |-> T ) )
49	42, 48	syl5eq 2125	. 2 |- ( ph -> ( w e. ( A X. B ) |-> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S ) = ( x e. A , y e. B |-> T ) )
50	31, 49	eqtrd 2113	1 |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   [_csb 2908   <.cop 3401   |-> cmpt 3839   X. cxp 4361   o. ccom 4367   -->wf 4918   ` cfv 4922   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788

This theorem is referenced by:  oprabco  5858