Theorem fmptcos 5353

Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

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 ) )

Assertion

Ref	Expression
fmptcos	|- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )

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

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

Proof of Theorem fmptcos

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptcof.1	. 2 |- ( ph -> A. x e. A R e. B )
2		fmptcof.2	. 2 |- ( ph -> F = ( x e. A |-> R ) )
3		fmptcof.3	. . 3 |- ( ph -> G = ( y e. B |-> S ) )
4		nfcv 2219	. . . 4 |- F/_ z S
5		nfcsb1v 2938	. . . 4 |- F/_ y [_ z / y ]_ S
6		csbeq1a 2916	. . . 4 |- ( y = z -> S = [_ z / y ]_ S )
7	4, 5, 6	cbvmpt 3872	. . 3 |- ( y e. B |-> S ) = ( z e. B |-> [_ z / y ]_ S )
8	3, 7	syl6eq 2129	. 2 |- ( ph -> G = ( z e. B |-> [_ z / y ]_ S ) )
9		csbeq1 2911	. 2 |- ( z = R -> [_ z / y ]_ S = [_ R / y ]_ S )
10	1, 2, 8, 9	fmptcof 5352	1 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )

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:  fmpt2co  5857