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Theorem oprabco 6864
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 466 . . 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 11 . . 3  |-  ( H  Fn  D  ->  F  =  ( x  e.  A ,  y  e.  B  |->  C ) )
5 dffn5 5911 . . . 4  |-  ( H  Fn  D  <->  H  =  ( z  e.  D  |->  ( H `  z
) ) )
65biimpi 194 . . 3  |-  ( H  Fn  D  ->  H  =  ( z  e.  D  |->  ( H `  z ) ) )
7 fveq2 5864 . . 3  |-  ( z  =  C  ->  ( H `  z )  =  ( H `  C ) )
82, 4, 6, 7fmpt2co 6863 . 2  |-  ( H  Fn  D  ->  ( H  o.  F )  =  ( x  e.  A ,  y  e.  B  |->  ( H `  C ) ) )
9 oprabco.3 . 2  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( H `  C
) )
108, 9syl6reqr 2527 1  |-  ( H  Fn  D  ->  G  =  ( H  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505    o. ccom 5003    Fn wfn 5581   ` cfv 5586    |-> cmpt2 6284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782
This theorem is referenced by:  oprab2co  6865
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