Theorem fmptcos 5634

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 2299	. . . 4 |- F/_ z S
5		nfcsb1v 3064	. . . 4 |- F/_ y [_ z / y ]_ S
6		csbeq1a 3040	. . . 4 |- ( y = z -> S = [_ z / y ]_ S )
7	4, 5, 6	cbvmpt 4059	. . 3 |- ( y e. B |-> S ) = ( z e. B |-> [_ z / y ]_ S )
8	3, 7	eqtrdi 2206	. 2 |- ( ph -> G = ( z e. B |-> [_ z / y ]_ S ) )
9		csbeq1 3034	. 2 |- ( z = R -> [_ z / y ]_ S = [_ R / y ]_ S )
10	1, 2, 8, 9	fmptcof 5633	1 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   A.wral 2435   [_csb 3031   |-> cmpt 4025   o. ccom 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177

This theorem is referenced by:  fmpoco  6160