Theorem fmptcos 5815

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 2374	. . . 4 |- F/_ z S
5		nfcsb1v 3160	. . . 4 |- F/_ y [_ z / y ]_ S
6		csbeq1a 3136	. . . 4 |- ( y = z -> S = [_ z / y ]_ S )
7	4, 5, 6	cbvmpt 4184	. . 3 |- ( y e. B |-> S ) = ( z e. B |-> [_ z / y ]_ S )
8	3, 7	eqtrdi 2280	. 2 |- ( ph -> G = ( z e. B |-> [_ z / y ]_ S ) )
9		csbeq1 3130	. 2 |- ( z = R -> [_ z / y ]_ S = [_ R / y ]_ S )
10	1, 2, 8, 9	fmptcof 5814	1 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   A.wral 2510   [_csb 3127   |-> cmpt 4150   o. ccom 4729

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by:  fmpoco  6380  divcncfap  15337