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Theorem fmptcos 5905
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.
StepHypRef 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 2574 . . . 4  |-  F/_ z S
5 nfcsb1v 3285 . . . 4  |-  F/_ y [_ z  /  y ]_ S
6 csbeq1a 3261 . . . 4  |-  ( y  =  z  ->  S  =  [_ z  /  y ]_ S )
74, 5, 6cbvmpt 4301 . . 3  |-  ( y  e.  B  |->  S )  =  ( z  e.  B  |->  [_ z  /  y ]_ S )
83, 7syl6eq 2486 . 2  |-  ( ph  ->  G  =  ( z  e.  B  |->  [_ z  /  y ]_ S
) )
9 csbeq1 3256 . 2  |-  ( z  =  R  ->  [_ z  /  y ]_ S  =  [_ R  /  y ]_ S )
101, 2, 8, 9fmptcof 5904 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   [_csb 3253    e. cmpt 4268    o. ccom 4884
This theorem is referenced by:  fmpt2co  6432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464
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