Theorem fmptcof 5331

Description: Version of fmptco 5330 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 2882	. . . . . . 7 |- F/_ x [_ z / x ]_ R
3	2	nfel1 2188	. . . . . 6 |- F/ x [_ z / x ]_ R e. B
4		csbeq1a 2860	. . . . . . 7 |- ( x = z -> R = [_ z / x ]_ R )
5	4	eleq1d 2106	. . . . . 6 |- ( x = z -> ( R e. B <-> [_ z / x ]_ R e. B ) )
6	3, 5	rspc 2650	. . . . 5 |- ( z e. A -> ( A. x e. A R e. B -> [_ z / x ]_ R e. B ) )
7	1, 6	mpan9 265	. . . 4 |- ( ( ph /\ z e. A ) -> [_ z / x ]_ R e. B )
8		fmptcof.2	. . . . 5 |- ( ph -> F = ( x e. A |-> R ) )
9		nfcv 2178	. . . . . 6 |- F/_ z R
10	9, 2, 4	cbvmpt 3851	. . . . 5 |- ( x e. A |-> R ) = ( z e. A |-> [_ z / x ]_ R )
11	8, 10	syl6eq 2088	. . . 4 |- ( ph -> F = ( z e. A |-> [_ z / x ]_ R ) )
12		fmptcof.3	. . . . 5 |- ( ph -> G = ( y e. B |-> S ) )
13		nfcv 2178	. . . . . 6 |- F/_ w S
14		nfcsb1v 2882	. . . . . 6 |- F/_ y [_ w / y ]_ S
15		csbeq1a 2860	. . . . . 6 |- ( y = w -> S = [_ w / y ]_ S )
16	13, 14, 15	cbvmpt 3851	. . . . 5 |- ( y e. B |-> S ) = ( w e. B |-> [_ w / y ]_ S )
17	12, 16	syl6eq 2088	. . . 4 |- ( ph -> G = ( w e. B |-> [_ w / y ]_ S ) )
18		csbeq1 2855	. . . 4 |- ( w = [_ z / x ]_ R -> [_ w / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
19	7, 11, 17, 18	fmptco 5330	. . 3 |- ( ph -> ( G o. F ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S ) )
20		nfcv 2178	. . . 4 |- F/_ z [_ R / y ]_ S
21		nfcv 2178	. . . . 5 |- F/_ x S
22	2, 21	nfcsb 2884	. . . 4 |- F/_ x [_ [_ z / x ]_ R / y ]_ S
23	4	csbeq1d 2858	. . . 4 |- ( x = z -> [_ R / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
24	20, 22, 23	cbvmpt 3851	. . 3 |- ( x e. A |-> [_ R / y ]_ S ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S )
25	19, 24	syl6eqr 2090	. 2 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )
26		eqid 2040	. . . 4 |- A = A
27		nfcvd 2179	. . . . . 6 |- ( R e. B -> F/_ y T )
28		fmptcof.4	. . . . . 6 |- ( y = R -> S = T )
29	27, 28	csbiegf 2890	. . . . 5 |- ( R e. B -> [_ R / y ]_ S = T )
30	29	ralimi 2384	. . . 4 |- ( A. x e. A R e. B -> A. x e. A [_ R / y ]_ S = T )
31		mpteq12 3840	. . . 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 393	. . 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 2072	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   A.wral 2306   [_csb 2852   |-> cmpt 3818   o. ccom 4349

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-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

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-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

This theorem is referenced by:  fmptcos  5332