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Theorem fmptcos 5697
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 2329 . . . 4  |-  F/_ z S
5 nfcsb1v 3102 . . . 4  |-  F/_ y [_ z  /  y ]_ S
6 csbeq1a 3078 . . . 4  |-  ( y  =  z  ->  S  =  [_ z  /  y ]_ S )
74, 5, 6cbvmpt 4110 . . 3  |-  ( y  e.  B  |->  S )  =  ( z  e.  B  |->  [_ z  /  y ]_ S )
83, 7eqtrdi 2236 . 2  |-  ( ph  ->  G  =  ( z  e.  B  |->  [_ z  /  y ]_ S
) )
9 csbeq1 3072 . 2  |-  ( z  =  R  ->  [_ z  /  y ]_ S  =  [_ R  /  y ]_ S )
101, 2, 8, 9fmptcof 5696 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158   A.wral 2465   [_csb 3069    |-> cmpt 4076    o. ccom 4642
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236
This theorem is referenced by:  fmpoco  6231
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