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Theorem oprabco 5920
Description: Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Hypotheses
Ref Expression
oprabco.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  D )
oprabco.2  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
oprabco.3  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( H `  C
) )
Assertion
Ref Expression
oprabco  |-  ( H  Fn  D  ->  G  =  ( H  o.  F ) )
Distinct variable groups:    x, y, A   
x, B, y    x, D, y    x, H, y
Allowed substitution hints:    C( x, y)    F( x, y)    G( x, y)

Proof of Theorem oprabco
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 oprabco.1 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  D )
21adantl 271 . . 3  |-  ( ( H  Fn  D  /\  ( x  e.  A  /\  y  e.  B
) )  ->  C  e.  D )
3 oprabco.2 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
43a1i 9 . . 3  |-  ( H  Fn  D  ->  F  =  ( x  e.  A ,  y  e.  B  |->  C ) )
5 dffn5im 5298 . . 3  |-  ( H  Fn  D  ->  H  =  ( z  e.  D  |->  ( H `  z ) ) )
6 fveq2 5256 . . 3  |-  ( z  =  C  ->  ( H `  z )  =  ( H `  C ) )
72, 4, 5, 6fmpt2co 5919 . 2  |-  ( H  Fn  D  ->  ( H  o.  F )  =  ( x  e.  A ,  y  e.  B  |->  ( H `  C ) ) )
8 oprabco.3 . 2  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( H `  C
) )
97, 8syl6reqr 2136 1  |-  ( H  Fn  D  ->  G  =  ( H  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436    o. ccom 4408    Fn wfn 4967   ` cfv 4972    |-> cmpt2 5596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fv 4980  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850
This theorem is referenced by:  oprab2co  5921
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