Theorem fmpt2co 5837

Description: Composition of two functions. Variation of fmptco 5330 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 u v 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 2401	. . . . 5 |- ( ph -> A. x e. A A. y e. B R e. C )
3		eqid 2040	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( x e. A , y e. B |-> R )
4	3	fmpt2 5827	. . . . 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 127	. . . 4 |- ( ph -> ( x e. A , y e. B |-> R ) : ( A X. B ) --> C )
6		nfcv 2178	. . . . . . 7 |- F/_ u R
7		nfcv 2178	. . . . . . 7 |- F/_ v R
8		nfcv 2178	. . . . . . . 8 |- F/_ x v
9		nfcsb1v 2882	. . . . . . . 8 |- F/_ x [_ u / x ]_ R
10	8, 9	nfcsb 2884	. . . . . . 7 |- F/_ x [_ v / y ]_ [_ u / x ]_ R
11		nfcsb1v 2882	. . . . . . 7 |- F/_ y [_ v / y ]_ [_ u / x ]_ R
12		csbeq1a 2860	. . . . . . . 8 |- ( x = u -> R = [_ u / x ]_ R )
13		csbeq1a 2860	. . . . . . . 8 |- ( y = v -> [_ u / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
14	12, 13	sylan9eq 2092	. . . . . . 7 |- ( ( x = u /\ y = v ) -> R = [_ v / y ]_ [_ u / x ]_ R )
15	6, 7, 10, 11, 14	cbvmpt2 5583	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( u e. A , v e. B |-> [_ v / y ]_ [_ u / x ]_ R )
16		vex 2560	. . . . . . . . . 10 |- u e. _V
17		vex 2560	. . . . . . . . . 10 |- v e. _V
18	16, 17	op2ndd 5776	. . . . . . . . 9 |- ( w = <. u , v >. -> ( 2nd ` w ) = v )
19	18	csbeq1d 2858	. . . . . . . 8 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R )
20	16, 17	op1std 5775	. . . . . . . . . 10 |- ( w = <. u , v >. -> ( 1st ` w ) = u )
21	20	csbeq1d 2858	. . . . . . . . 9 |- ( w = <. u , v >. -> [_ ( 1st ` w ) / x ]_ R = [_ u / x ]_ R )
22	21	csbeq2dv 2875	. . . . . . . 8 |- ( w = <. u , v >. -> [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
23	19, 22	eqtrd 2072	. . . . . . 7 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
24	23	mpt2mpt 5596	. . . . . 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 2063	. . . . 5 |- ( x e. A , y e. B |-> R ) = ( w e. ( A X. B ) |-> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R )
26	25	fmpt 5319	. . . 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 137	. . 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 2088	. . 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 5332	. 2 |- ( ph -> ( G o. F ) = ( w e. ( A X. B ) |-> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S ) )
32	23	csbeq1d 2858	. . . . 5 |- ( w = <. u , v >. -> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
33	32	mpt2mpt 5596	. . . 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 2178	. . . . 5 |- F/_ u [_ R / z ]_ S
35		nfcv 2178	. . . . 5 |- F/_ v [_ R / z ]_ S
36		nfcv 2178	. . . . . 6 |- F/_ x S
37	10, 36	nfcsb 2884	. . . . 5 |- F/_ x [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
38		nfcv 2178	. . . . . 6 |- F/_ y S
39	11, 38	nfcsb 2884	. . . . 5 |- F/_ y [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
40	14	csbeq1d 2858	. . . . 5 |- ( ( x = u /\ y = v ) -> [_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
41	34, 35, 37, 39, 40	cbvmpt2 5583	. . . 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 2063	. . 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 1100	. . . . 5 |- ( ( ph /\ x e. A /\ y e. B ) -> R e. C )
44		nfcvd 2179	. . . . . 6 |- ( R e. C -> F/_ z T )
45		fmpt2co.4	. . . . . 6 |- ( z = R -> S = T )
46	44, 45	csbiegf 2890	. . . . 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 5569	. . 3 |- ( ph -> ( x e. A , y e. B |-> [_ R / z ]_ S ) = ( x e. A , y e. B |-> T ) )
49	42, 48	syl5eq 2084	. 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 2072	1 |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   A.wral 2306   [_csb 2852   <.cop 3378   |-> cmpt 3818   X. cxp 4343   o. ccom 4349   -->wf 4898   ` cfv 4902   |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768

This theorem is referenced by:  oprabco  5838