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Theorem fmptcos 5980
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 2613 . . . 4  |-  F/_ z S
5 nfcsb1v 3405 . . . 4  |-  F/_ y [_ z  /  y ]_ S
6 csbeq1a 3398 . . . 4  |-  ( y  =  z  ->  S  =  [_ z  /  y ]_ S )
74, 5, 6cbvmpt 4483 . . 3  |-  ( y  e.  B  |->  S )  =  ( z  e.  B  |->  [_ z  /  y ]_ S )
83, 7syl6eq 2508 . 2  |-  ( ph  ->  G  =  ( z  e.  B  |->  [_ z  /  y ]_ S
) )
9 csbeq1 3392 . 2  |-  ( z  =  R  ->  [_ z  /  y ]_ S  =  [_ R  /  y ]_ S )
101, 2, 8, 9fmptcof 5979 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2795   [_csb 3389    |-> cmpt 4451    o. ccom 4945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527
This theorem is referenced by:  fmpt2co  6759  gsummptshft  16543  gsummptf1o  26385
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