Theorem fmpoco 6065

Description: Composition of two functions. Variation of fmptco 5538 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypotheses

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

Assertion

Ref	Expression
fmpoco	|- ( 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 fmpoco

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

Step	Hyp	Ref	Expression
1		fmpoco.1	. . . . . 6 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> R e. C )
2	1	ralrimivva 2486	. . . . 5 |- ( ph -> A. x e. A A. y e. B R e. C )
3		eqid 2113	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( x e. A , y e. B |-> R )
4	3	fmpo 6051	. . . . 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 121	. . . 4 |- ( ph -> ( x e. A , y e. B |-> R ) : ( A X. B ) --> C )
6		nfcv 2253	. . . . . . 7 |- F/_ u R
7		nfcv 2253	. . . . . . 7 |- F/_ v R
8		nfcv 2253	. . . . . . . 8 |- F/_ x v
9		nfcsb1v 2999	. . . . . . . 8 |- F/_ x [_ u / x ]_ R
10	8, 9	nfcsb 3001	. . . . . . 7 |- F/_ x [_ v / y ]_ [_ u / x ]_ R
11		nfcsb1v 2999	. . . . . . 7 |- F/_ y [_ v / y ]_ [_ u / x ]_ R
12		csbeq1a 2977	. . . . . . . 8 |- ( x = u -> R = [_ u / x ]_ R )
13		csbeq1a 2977	. . . . . . . 8 |- ( y = v -> [_ u / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
14	12, 13	sylan9eq 2165	. . . . . . 7 |- ( ( x = u /\ y = v ) -> R = [_ v / y ]_ [_ u / x ]_ R )
15	6, 7, 10, 11, 14	cbvmpo 5802	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( u e. A , v e. B |-> [_ v / y ]_ [_ u / x ]_ R )
16		vex 2658	. . . . . . . . . 10 |- u e. _V
17		vex 2658	. . . . . . . . . 10 |- v e. _V
18	16, 17	op2ndd 5999	. . . . . . . . 9 |- ( w = <. u , v >. -> ( 2nd ` w ) = v )
19	18	csbeq1d 2975	. . . . . . . 8 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R )
20	16, 17	op1std 5998	. . . . . . . . . 10 |- ( w = <. u , v >. -> ( 1st ` w ) = u )
21	20	csbeq1d 2975	. . . . . . . . 9 |- ( w = <. u , v >. -> [_ ( 1st ` w ) / x ]_ R = [_ u / x ]_ R )
22	21	csbeq2dv 2992	. . . . . . . 8 |- ( w = <. u , v >. -> [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
23	19, 22	eqtrd 2145	. . . . . . 7 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
24	23	mpompt 5815	. . . . . 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 2136	. . . . 5 |- ( x e. A , y e. B |-> R ) = ( w e. ( A X. B ) |-> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R )
26	25	fmpt 5522	. . . 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 133	. . 3 |- ( ph -> A. w e. ( A X. B ) [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R e. C )
28		fmpoco.2	. . . 4 |- ( ph -> F = ( x e. A , y e. B |-> R ) )
29	28, 25	syl6eq 2161	. . 3 |- ( ph -> F = ( w e. ( A X. B ) |-> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R ) )
30		fmpoco.3	. . 3 |- ( ph -> G = ( z e. C |-> S ) )
31	27, 29, 30	fmptcos 5540	. 2 |- ( ph -> ( G o. F ) = ( w e. ( A X. B ) |-> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S ) )
32	23	csbeq1d 2975	. . . . 5 |- ( w = <. u , v >. -> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
33	32	mpompt 5815	. . . 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 2253	. . . . 5 |- F/_ u [_ R / z ]_ S
35		nfcv 2253	. . . . 5 |- F/_ v [_ R / z ]_ S
36		nfcv 2253	. . . . . 6 |- F/_ x S
37	10, 36	nfcsb 3001	. . . . 5 |- F/_ x [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
38		nfcv 2253	. . . . . 6 |- F/_ y S
39	11, 38	nfcsb 3001	. . . . 5 |- F/_ y [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
40	14	csbeq1d 2975	. . . . 5 |- ( ( x = u /\ y = v ) -> [_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
41	34, 35, 37, 39, 40	cbvmpo 5802	. . . 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 2136	. . 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 1158	. . . . 5 |- ( ( ph /\ x e. A /\ y e. B ) -> R e. C )
44		nfcvd 2254	. . . . . 6 |- ( R e. C -> F/_ z T )
45		fmpoco.4	. . . . . 6 |- ( z = R -> S = T )
46	44, 45	csbiegf 3007	. . . . 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	mpoeq3dva 5787	. . 3 |- ( ph -> ( x e. A , y e. B |-> [_ R / z ]_ S ) = ( x e. A , y e. B |-> T ) )
49	42, 48	syl5eq 2157	. 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 2145	1 |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 943   = wceq 1312   e. wcel 1461   A.wral 2388   [_csb 2969   <.cop 3494   |-> cmpt 3947   X. cxp 4495   o. ccom 4501   -->wf 5075   ` cfv 5079   e. cmpo 5728   1stc1st 5988   2ndc2nd 5989

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991

This theorem is referenced by:  oprabco  6066  bdxmet  12484