Theorem fmpoco 6175

Description: Composition of two functions. Variation of fmptco 5645 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 2546	. . . . 5 |- ( ph -> A. x e. A A. y e. B R e. C )
3		eqid 2164	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( x e. A , y e. B |-> R )
4	3	fmpo 6161	. . . . 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 2306	. . . . . . 7 |- F/_ u R
7		nfcv 2306	. . . . . . 7 |- F/_ v R
8		nfcv 2306	. . . . . . . 8 |- F/_ x v
9		nfcsb1v 3073	. . . . . . . 8 |- F/_ x [_ u / x ]_ R
10	8, 9	nfcsb 3077	. . . . . . 7 |- F/_ x [_ v / y ]_ [_ u / x ]_ R
11		nfcsb1v 3073	. . . . . . 7 |- F/_ y [_ v / y ]_ [_ u / x ]_ R
12		csbeq1a 3049	. . . . . . . 8 |- ( x = u -> R = [_ u / x ]_ R )
13		csbeq1a 3049	. . . . . . . 8 |- ( y = v -> [_ u / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
14	12, 13	sylan9eq 2217	. . . . . . 7 |- ( ( x = u /\ y = v ) -> R = [_ v / y ]_ [_ u / x ]_ R )
15	6, 7, 10, 11, 14	cbvmpo 5912	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( u e. A , v e. B |-> [_ v / y ]_ [_ u / x ]_ R )
16		vex 2724	. . . . . . . . . 10 |- u e. _V
17		vex 2724	. . . . . . . . . 10 |- v e. _V
18	16, 17	op2ndd 6109	. . . . . . . . 9 |- ( w = <. u , v >. -> ( 2nd ` w ) = v )
19	18	csbeq1d 3047	. . . . . . . 8 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R )
20	16, 17	op1std 6108	. . . . . . . . . 10 |- ( w = <. u , v >. -> ( 1st ` w ) = u )
21	20	csbeq1d 3047	. . . . . . . . 9 |- ( w = <. u , v >. -> [_ ( 1st ` w ) / x ]_ R = [_ u / x ]_ R )
22	21	csbeq2dv 3066	. . . . . . . 8 |- ( w = <. u , v >. -> [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
23	19, 22	eqtrd 2197	. . . . . . 7 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
24	23	mpompt 5925	. . . . . 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 2188	. . . . 5 |- ( x e. A , y e. B |-> R ) = ( w e. ( A X. B ) |-> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R )
26	25	fmpt 5629	. . . 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	eqtrdi 2213	. . 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 5647	. 2 |- ( ph -> ( G o. F ) = ( w e. ( A X. B ) |-> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S ) )
32	23	csbeq1d 3047	. . . . 5 |- ( w = <. u , v >. -> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
33	32	mpompt 5925	. . . 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 2306	. . . . 5 |- F/_ u [_ R / z ]_ S
35		nfcv 2306	. . . . 5 |- F/_ v [_ R / z ]_ S
36		nfcv 2306	. . . . . 6 |- F/_ x S
37	10, 36	nfcsb 3077	. . . . 5 |- F/_ x [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
38		nfcv 2306	. . . . . 6 |- F/_ y S
39	11, 38	nfcsb 3077	. . . . 5 |- F/_ y [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
40	14	csbeq1d 3047	. . . . 5 |- ( ( x = u /\ y = v ) -> [_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
41	34, 35, 37, 39, 40	cbvmpo 5912	. . . 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 2188	. . 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 1188	. . . . 5 |- ( ( ph /\ x e. A /\ y e. B ) -> R e. C )
44		nfcvd 2307	. . . . . 6 |- ( R e. C -> F/_ z T )
45		fmpoco.4	. . . . . 6 |- ( z = R -> S = T )
46	44, 45	csbiegf 3083	. . . . 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 5897	. . 3 |- ( ph -> ( x e. A , y e. B |-> [_ R / z ]_ S ) = ( x e. A , y e. B |-> T ) )
49	42, 48	syl5eq 2209	. 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 2197	1 |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 967   = wceq 1342   e. wcel 2135   A.wral 2442   [_csb 3040   <.cop 3573   |-> cmpt 4037   X. cxp 4596   o. ccom 4602   -->wf 5178   ` cfv 5182   e. cmpo 5838   1stc1st 6098   2ndc2nd 6099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101

This theorem is referenced by:  oprabco  6176  txswaphmeolem  12861  bdxmet  13042