Theorem fmptcof 5352

Description: Version of fmptco 5351 where ph needn't be distinct from x. (Contributed by NM, 27-Dec-2014.)

Hypotheses

Ref	Expression
fmptcof.1	|- ( ph -> A. x e. A R e. B )
fmptcof.2	|- ( ph -> F = ( x e. A |-> R ) )
fmptcof.3	|- ( ph -> G = ( y e. B |-> S ) )
fmptcof.4	|- ( y = R -> S = T )

Assertion

Ref	Expression
fmptcof	|- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

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

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

Proof of Theorem fmptcof

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptcof.1	. . . . 5 |- ( ph -> A. x e. A R e. B )
2		nfcsb1v 2938	. . . . . . 7 |- F/_ x [_ z / x ]_ R
3	2	nfel1 2229	. . . . . 6 |- F/ x [_ z / x ]_ R e. B
4		csbeq1a 2916	. . . . . . 7 |- ( x = z -> R = [_ z / x ]_ R )
5	4	eleq1d 2147	. . . . . 6 |- ( x = z -> ( R e. B <-> [_ z / x ]_ R e. B ) )
6	3, 5	rspc 2695	. . . . 5 |- ( z e. A -> ( A. x e. A R e. B -> [_ z / x ]_ R e. B ) )
7	1, 6	mpan9 275	. . . 4 |- ( ( ph /\ z e. A ) -> [_ z / x ]_ R e. B )
8		fmptcof.2	. . . . 5 |- ( ph -> F = ( x e. A |-> R ) )
9		nfcv 2219	. . . . . 6 |- F/_ z R
10	9, 2, 4	cbvmpt 3872	. . . . 5 |- ( x e. A |-> R ) = ( z e. A |-> [_ z / x ]_ R )
11	8, 10	syl6eq 2129	. . . 4 |- ( ph -> F = ( z e. A |-> [_ z / x ]_ R ) )
12		fmptcof.3	. . . . 5 |- ( ph -> G = ( y e. B |-> S ) )
13		nfcv 2219	. . . . . 6 |- F/_ w S
14		nfcsb1v 2938	. . . . . 6 |- F/_ y [_ w / y ]_ S
15		csbeq1a 2916	. . . . . 6 |- ( y = w -> S = [_ w / y ]_ S )
16	13, 14, 15	cbvmpt 3872	. . . . 5 |- ( y e. B |-> S ) = ( w e. B |-> [_ w / y ]_ S )
17	12, 16	syl6eq 2129	. . . 4 |- ( ph -> G = ( w e. B |-> [_ w / y ]_ S ) )
18		csbeq1 2911	. . . 4 |- ( w = [_ z / x ]_ R -> [_ w / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
19	7, 11, 17, 18	fmptco 5351	. . 3 |- ( ph -> ( G o. F ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S ) )
20		nfcv 2219	. . . 4 |- F/_ z [_ R / y ]_ S
21		nfcv 2219	. . . . 5 |- F/_ x S
22	2, 21	nfcsb 2940	. . . 4 |- F/_ x [_ [_ z / x ]_ R / y ]_ S
23	4	csbeq1d 2914	. . . 4 |- ( x = z -> [_ R / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
24	20, 22, 23	cbvmpt 3872	. . 3 |- ( x e. A |-> [_ R / y ]_ S ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S )
25	19, 24	syl6eqr 2131	. 2 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )
26		eqid 2081	. . . 4 |- A = A
27		nfcvd 2220	. . . . . 6 |- ( R e. B -> F/_ y T )
28		fmptcof.4	. . . . . 6 |- ( y = R -> S = T )
29	27, 28	csbiegf 2946	. . . . 5 |- ( R e. B -> [_ R / y ]_ S = T )
30	29	ralimi 2426	. . . 4 |- ( A. x e. A R e. B -> A. x e. A [_ R / y ]_ S = T )
31		mpteq12 3861	. . . 4 |- ( ( A = A /\ A. x e. A [_ R / y ]_ S = T ) -> ( x e. A |-> [_ R / y ]_ S ) = ( x e. A |-> T ) )
32	26, 30, 31	sylancr 405	. . 3 |- ( A. x e. A R e. B -> ( x e. A |-> [_ R / y ]_ S ) = ( x e. A |-> T ) )
33	1, 32	syl 14	. 2 |- ( ph -> ( x e. A |-> [_ R / y ]_ S ) = ( x e. A |-> T ) )
34	25, 33	eqtrd 2113	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   A.wral 2348   [_csb 2908   |-> cmpt 3839   o. ccom 4367

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-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

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-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

This theorem is referenced by:  fmptcos  5353