Theorem fmptcos 5845

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 2384	. . . 4 |- F/_ z S
5		nfcsb1v 3171	. . . 4 |- F/_ y [_ z / y ]_ S
6		csbeq1a 3147	. . . 4 |- ( y = z -> S = [_ z / y ]_ S )
7	4, 5, 6	cbvmpt 4205	. . 3 |- ( y e. B |-> S ) = ( z e. B |-> [_ z / y ]_ S )
8	3, 7	eqtrdi 2281	. 2 |- ( ph -> G = ( z e. B |-> [_ z / y ]_ S ) )
9		csbeq1 3141	. 2 |- ( z = R -> [_ z / y ]_ S = [_ R / y ]_ S )
10	1, 2, 8, 9	fmptcof 5844	1 |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   A.wral 2520   [_csb 3138   |-> cmpt 4171   o. ccom 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360

This theorem is referenced by:  fmpoco  6412  divcncfap  15479