Theorem fmpt2co 7260

Description: Composition of two functions. Variation of fmptco 6396 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 2971	. . . . 5 |- ( ph -> A. x e. A A. y e. B R e. C )
3		eqid 2622	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( x e. A , y e. B |-> R )
4	3	fmpt2 7237	. . . . 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 208	. . . 4 |- ( ph -> ( x e. A , y e. B |-> R ) : ( A X. B ) --> C )
6		nfcv 2764	. . . . . . 7 |- F/_ u R
7		nfcv 2764	. . . . . . 7 |- F/_ v R
8		nfcv 2764	. . . . . . . 8 |- F/_ x v
9		nfcsb1v 3549	. . . . . . . 8 |- F/_ x [_ u / x ]_ R
10	8, 9	nfcsb 3551	. . . . . . 7 |- F/_ x [_ v / y ]_ [_ u / x ]_ R
11		nfcsb1v 3549	. . . . . . 7 |- F/_ y [_ v / y ]_ [_ u / x ]_ R
12		csbeq1a 3542	. . . . . . . 8 |- ( x = u -> R = [_ u / x ]_ R )
13		csbeq1a 3542	. . . . . . . 8 |- ( y = v -> [_ u / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
14	12, 13	sylan9eq 2676	. . . . . . 7 |- ( ( x = u /\ y = v ) -> R = [_ v / y ]_ [_ u / x ]_ R )
15	6, 7, 10, 11, 14	cbvmpt2 6734	. . . . . 6 |- ( x e. A , y e. B |-> R ) = ( u e. A , v e. B |-> [_ v / y ]_ [_ u / x ]_ R )
16		vex 3203	. . . . . . . . . 10 |- u e. _V
17		vex 3203	. . . . . . . . . 10 |- v e. _V
18	16, 17	op2ndd 7179	. . . . . . . . 9 |- ( w = <. u , v >. -> ( 2nd ` w ) = v )
19	18	csbeq1d 3540	. . . . . . . 8 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R )
20	16, 17	op1std 7178	. . . . . . . . . 10 |- ( w = <. u , v >. -> ( 1st ` w ) = u )
21	20	csbeq1d 3540	. . . . . . . . 9 |- ( w = <. u , v >. -> [_ ( 1st ` w ) / x ]_ R = [_ u / x ]_ R )
22	21	csbeq2dv 3992	. . . . . . . 8 |- ( w = <. u , v >. -> [_ v / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
23	19, 22	eqtrd 2656	. . . . . . 7 |- ( w = <. u , v >. -> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R = [_ v / y ]_ [_ u / x ]_ R )
24	23	mpt2mpt 6752	. . . . . 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 2647	. . . . 5 |- ( x e. A , y e. B |-> R ) = ( w e. ( A X. B ) |-> [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R )
26	25	fmpt 6381	. . . 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 224	. . 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 2672	. . 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 6398	. 2 |- ( ph -> ( G o. F ) = ( w e. ( A X. B ) |-> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S ) )
32	23	csbeq1d 3540	. . . . 5 |- ( w = <. u , v >. -> [_ [_ ( 2nd ` w ) / y ]_ [_ ( 1st ` w ) / x ]_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
33	32	mpt2mpt 6752	. . . 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 2764	. . . . 5 |- F/_ u [_ R / z ]_ S
35		nfcv 2764	. . . . 5 |- F/_ v [_ R / z ]_ S
36		nfcv 2764	. . . . . 6 |- F/_ x S
37	10, 36	nfcsb 3551	. . . . 5 |- F/_ x [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
38		nfcv 2764	. . . . . 6 |- F/_ y S
39	11, 38	nfcsb 3551	. . . . 5 |- F/_ y [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S
40	14	csbeq1d 3540	. . . . 5 |- ( ( x = u /\ y = v ) -> [_ R / z ]_ S = [_ [_ v / y ]_ [_ u / x ]_ R / z ]_ S )
41	34, 35, 37, 39, 40	cbvmpt2 6734	. . . 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 2647	. . 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 1260	. . . . 5 |- ( ( ph /\ x e. A /\ y e. B ) -> R e. C )
44		nfcvd 2765	. . . . . 6 |- ( R e. C -> F/_ z T )
45		fmpt2co.4	. . . . . 6 |- ( z = R -> S = T )
46	44, 45	csbiegf 3557	. . . . 5 |- ( R e. C -> [_ R / z ]_ S = T )
47	43, 46	syl 17	. . . 4 |- ( ( ph /\ x e. A /\ y e. B ) -> [_ R / z ]_ S = T )
48	47	mpt2eq3dva 6719	. . 3 |- ( ph -> ( x e. A , y e. B |-> [_ R / z ]_ S ) = ( x e. A , y e. B |-> T ) )
49	42, 48	syl5eq 2668	. 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 2656	1 |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   [_csb 3533   <.cop 4183   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888   |-> cmpt2 6652   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-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-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-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-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-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  oprabco  7261  evlslem2  19512  txswaphmeolem  21607  xpstopnlem1  21612  stdbdxmet  22320  rrxds  23181  cnre2csqima  29957  cvmlift2lem7  31291