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Theorem fmptcos 5556
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 2258 . . . 4  |-  F/_ z S
5 nfcsb1v 3005 . . . 4  |-  F/_ y [_ z  /  y ]_ S
6 csbeq1a 2983 . . . 4  |-  ( y  =  z  ->  S  =  [_ z  /  y ]_ S )
74, 5, 6cbvmpt 3993 . . 3  |-  ( y  e.  B  |->  S )  =  ( z  e.  B  |->  [_ z  /  y ]_ S )
83, 7syl6eq 2166 . 2  |-  ( ph  ->  G  =  ( z  e.  B  |->  [_ z  /  y ]_ S
) )
9 csbeq1 2978 . 2  |-  ( z  =  R  ->  [_ z  /  y ]_ S  =  [_ R  /  y ]_ S )
101, 2, 8, 9fmptcof 5555 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   A.wral 2393   [_csb 2975    |-> cmpt 3959    o. ccom 4513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by:  fmpoco  6081
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