Theorem fmptcos 5332

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 2178	. . . 4 |- F/_ z S
5		nfcsb1v 2882	. . . 4 |- F/_ y [_ z / y ]_ S
6		csbeq1a 2860	. . . 4 |- ( y = z -> S = [_ z / y ]_ S )
7	4, 5, 6	cbvmpt 3851	. . 3 |- ( y e. B |-> S ) = ( z e. B |-> [_ z / y ]_ S )
8	3, 7	syl6eq 2088	. 2 |- ( ph -> G = ( z e. B |-> [_ z / y ]_ S ) )
9		csbeq1 2855	. 2 |- ( z = R -> [_ z / y ]_ S = [_ R / y ]_ S )
10	1, 2, 8, 9	fmptcof 5331	1 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   A.wral 2306   [_csb 2852   |-> cmpt 3818   o. ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910

This theorem is referenced by:  fmpt2co  5837