Theorem fmptcof 5810

Description: Version of fmptco 5809 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 3158	. . . . . . 7 |- F/_ x [_ z / x ]_ R
3	2	nfel1 2383	. . . . . 6 |- F/ x [_ z / x ]_ R e. B
4		csbeq1a 3134	. . . . . . 7 |- ( x = z -> R = [_ z / x ]_ R )
5	4	eleq1d 2298	. . . . . 6 |- ( x = z -> ( R e. B <-> [_ z / x ]_ R e. B ) )
6	3, 5	rspc 2902	. . . . 5 |- ( z e. A -> ( A. x e. A R e. B -> [_ z / x ]_ R e. B ) )
7	1, 6	mpan9 281	. . . 4 |- ( ( ph /\ z e. A ) -> [_ z / x ]_ R e. B )
8		fmptcof.2	. . . . 5 |- ( ph -> F = ( x e. A |-> R ) )
9		nfcv 2372	. . . . . 6 |- F/_ z R
10	9, 2, 4	cbvmpt 4182	. . . . 5 |- ( x e. A |-> R ) = ( z e. A |-> [_ z / x ]_ R )
11	8, 10	eqtrdi 2278	. . . 4 |- ( ph -> F = ( z e. A |-> [_ z / x ]_ R ) )
12		fmptcof.3	. . . . 5 |- ( ph -> G = ( y e. B |-> S ) )
13		nfcv 2372	. . . . . 6 |- F/_ w S
14		nfcsb1v 3158	. . . . . 6 |- F/_ y [_ w / y ]_ S
15		csbeq1a 3134	. . . . . 6 |- ( y = w -> S = [_ w / y ]_ S )
16	13, 14, 15	cbvmpt 4182	. . . . 5 |- ( y e. B |-> S ) = ( w e. B |-> [_ w / y ]_ S )
17	12, 16	eqtrdi 2278	. . . 4 |- ( ph -> G = ( w e. B |-> [_ w / y ]_ S ) )
18		csbeq1 3128	. . . 4 |- ( w = [_ z / x ]_ R -> [_ w / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
19	7, 11, 17, 18	fmptco 5809	. . 3 |- ( ph -> ( G o. F ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S ) )
20		nfcv 2372	. . . 4 |- F/_ z [_ R / y ]_ S
21		nfcv 2372	. . . . 5 |- F/_ x S
22	2, 21	nfcsb 3163	. . . 4 |- F/_ x [_ [_ z / x ]_ R / y ]_ S
23	4	csbeq1d 3132	. . . 4 |- ( x = z -> [_ R / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
24	20, 22, 23	cbvmpt 4182	. . 3 |- ( x e. A |-> [_ R / y ]_ S ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S )
25	19, 24	eqtr4di 2280	. 2 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )
26		eqid 2229	. . . 4 |- A = A
27		nfcvd 2373	. . . . . 6 |- ( R e. B -> F/_ y T )
28		fmptcof.4	. . . . . 6 |- ( y = R -> S = T )
29	27, 28	csbiegf 3169	. . . . 5 |- ( R e. B -> [_ R / y ]_ S = T )
30	29	ralimi 2593	. . . 4 |- ( A. x e. A R e. B -> A. x e. A [_ R / y ]_ S = T )
31		mpteq12 4170	. . . 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 414	. . 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 2262	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   A.wral 2508   [_csb 3125   |-> cmpt 4148   o. ccom 4727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332

This theorem is referenced by:  fmptcos  5811  cncfmpt1f  15312  sincn  15483  coscn  15484  lgseisenlem3  15791