Theorem fmptcof 5725

Description: Version of fmptco 5724 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 3113	. . . . . . 7 |- F/_ x [_ z / x ]_ R
3	2	nfel1 2347	. . . . . 6 |- F/ x [_ z / x ]_ R e. B
4		csbeq1a 3089	. . . . . . 7 |- ( x = z -> R = [_ z / x ]_ R )
5	4	eleq1d 2262	. . . . . 6 |- ( x = z -> ( R e. B <-> [_ z / x ]_ R e. B ) )
6	3, 5	rspc 2858	. . . . 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 2336	. . . . . 6 |- F/_ z R
10	9, 2, 4	cbvmpt 4124	. . . . 5 |- ( x e. A |-> R ) = ( z e. A |-> [_ z / x ]_ R )
11	8, 10	eqtrdi 2242	. . . 4 |- ( ph -> F = ( z e. A |-> [_ z / x ]_ R ) )
12		fmptcof.3	. . . . 5 |- ( ph -> G = ( y e. B |-> S ) )
13		nfcv 2336	. . . . . 6 |- F/_ w S
14		nfcsb1v 3113	. . . . . 6 |- F/_ y [_ w / y ]_ S
15		csbeq1a 3089	. . . . . 6 |- ( y = w -> S = [_ w / y ]_ S )
16	13, 14, 15	cbvmpt 4124	. . . . 5 |- ( y e. B |-> S ) = ( w e. B |-> [_ w / y ]_ S )
17	12, 16	eqtrdi 2242	. . . 4 |- ( ph -> G = ( w e. B |-> [_ w / y ]_ S ) )
18		csbeq1 3083	. . . 4 |- ( w = [_ z / x ]_ R -> [_ w / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
19	7, 11, 17, 18	fmptco 5724	. . 3 |- ( ph -> ( G o. F ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S ) )
20		nfcv 2336	. . . 4 |- F/_ z [_ R / y ]_ S
21		nfcv 2336	. . . . 5 |- F/_ x S
22	2, 21	nfcsb 3118	. . . 4 |- F/_ x [_ [_ z / x ]_ R / y ]_ S
23	4	csbeq1d 3087	. . . 4 |- ( x = z -> [_ R / y ]_ S = [_ [_ z / x ]_ R / y ]_ S )
24	20, 22, 23	cbvmpt 4124	. . 3 |- ( x e. A |-> [_ R / y ]_ S ) = ( z e. A |-> [_ [_ z / x ]_ R / y ]_ S )
25	19, 24	eqtr4di 2244	. 2 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )
26		eqid 2193	. . . 4 |- A = A
27		nfcvd 2337	. . . . . 6 |- ( R e. B -> F/_ y T )
28		fmptcof.4	. . . . . 6 |- ( y = R -> S = T )
29	27, 28	csbiegf 3124	. . . . 5 |- ( R e. B -> [_ R / y ]_ S = T )
30	29	ralimi 2557	. . . 4 |- ( A. x e. A R e. B -> A. x e. A [_ R / y ]_ S = T )
31		mpteq12 4112	. . . 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 2226	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   A.wral 2472   [_csb 3080   |-> cmpt 4090   o. ccom 4663

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262

This theorem is referenced by:  fmptcos  5726  cncfmpt1f  14752  sincn  14904  coscn  14905  lgseisenlem3  15188