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Theorem oprabco 6203
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 452 . . 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 10 . . 3  |-  ( H  Fn  D  ->  F  =  ( x  e.  A ,  y  e.  B  |->  C ) )
5 dffn5 5568 . . . 4  |-  ( H  Fn  D  <->  H  =  ( z  e.  D  |->  ( H `  z
) ) )
65biimpi 186 . . 3  |-  ( H  Fn  D  ->  H  =  ( z  e.  D  |->  ( H `  z ) ) )
7 fveq2 5525 . . 3  |-  ( z  =  C  ->  ( H `  z )  =  ( H `  C ) )
82, 4, 6, 7fmpt2co 6202 . 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 2334 1  |-  ( H  Fn  D  ->  G  =  ( H  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077    o. ccom 4693    Fn wfn 5250   ` cfv 5255    e. cmpt2 5860
This theorem is referenced by:  oprab2co  6204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
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