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Theorem oprabco 6760
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 5839 . . . 4  |-  ( H  Fn  D  <->  H  =  ( z  e.  D  |->  ( H `  z
) ) )
65biimpi 194 . . 3  |-  ( H  Fn  D  ->  H  =  ( z  e.  D  |->  ( H `  z ) ) )
7 fveq2 5792 . . 3  |-  ( z  =  C  ->  ( H `  z )  =  ( H `  C ) )
82, 4, 6, 7fmpt2co 6759 . 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 2511 1  |-  ( H  Fn  D  ->  G  =  ( H  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    |-> cmpt 4451    o. ccom 4945    Fn wfn 5514   ` cfv 5519    |-> cmpt2 6195
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  ax-un 6475
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-iun 4274  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  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681
This theorem is referenced by:  oprab2co  6761
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