Theorem fmptcof 5749

Description: Version of fmptco 5748 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 3126	. . . . . . 7 |- F/_ x [_ z / x ]_ R
3	2	nfel1 2359	. . . . . 6 |- F/ x [_ z / x ]_ R e. B
4		csbeq1a 3102	. . . . . . 7 |- ( x = z -> R = [_ z / x ]_ R )
5	4	eleq1d 2274	. . . . . 6 |- ( x = z -> ( R e. B <-> [_ z / x ]_ R e. B ) )
6	3, 5	rspc 2871	. . . . 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 2348	. . . . . 6 |- F/_ z R
10	9, 2, 4	cbvmpt 4140	. . . . 5 |- ( x e. A |-> R ) = ( z e. A |-> [_ z / x ]_ R )
11	8, 10	eqtrdi 2254	. . . 4 |- ( ph -> F = ( z e. A |-> [_ z / x ]_ R ) )
12		fmptcof.3	. . . . 5 |- ( ph -> G = ( y e. B |-> S ) )
13		nfcv 2348	. . . . . 6 |- F/_ w S
14		nfcsb1v 3126	. . . . . 6 |- F/_ y [_ w / y ]_ S
15		csbeq1a 3102	. . . . . 6 |- ( y = w -> S = [_ w / y ]_ S )
16	13, 14, 15	cbvmpt 4140	. . . . 5 |- ( y e. B |-> S ) = ( w e. B |-> [_ w / y ]_ S )
17	12, 16	eqtrdi 2254	. . . 4 |- ( ph -> G = ( w e. B |-> [_ w / y ]_ S ) )
18		csbeq1 3096	. . . 4 |- ( w = [_ z / x ]_ R -> [_ w / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
19	7, 11, 17, 18	fmptco 5748	. . 3 |- ( ph -> ( G o. F ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S ) )
20		nfcv 2348	. . . 4 |- F/_ z [_ R / y ]_ S
21		nfcv 2348	. . . . 5 |- F/_ x S
22	2, 21	nfcsb 3131	. . . 4 |- F/_ x [_ [_ z / x ]_ R / y ]_ S
23	4	csbeq1d 3100	. . . 4 |- ( x = z -> [_ R / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
24	20, 22, 23	cbvmpt 4140	. . 3 |- ( x e. A |-> [_ R / y ]_ S ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S )
25	19, 24	eqtr4di 2256	. 2 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )
26		eqid 2205	. . . 4 |- A = A
27		nfcvd 2349	. . . . . 6 |- ( R e. B -> F/_ y T )
28		fmptcof.4	. . . . . 6 |- ( y = R -> S = T )
29	27, 28	csbiegf 3137	. . . . 5 |- ( R e. B -> [_ R / y ]_ S = T )
30	29	ralimi 2569	. . . 4 |- ( A. x e. A R e. B -> A. x e. A [_ R / y ]_ S = T )
31		mpteq12 4128	. . . 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 2238	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   A.wral 2484   [_csb 3093   |-> cmpt 4106   o. ccom 4680

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280

This theorem is referenced by:  fmptcos  5750  cncfmpt1f  15103  sincn  15274  coscn  15275  lgseisenlem3  15582