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Theorem opropabco 26389
Description: Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
Hypotheses
Ref Expression
opropabco.1  |-  ( x  e.  A  ->  B  e.  R )
opropabco.2  |-  ( x  e.  A  ->  C  e.  S )
opropabco.3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  <. B ,  C >. ) }
opropabco.4  |-  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( B M C ) ) }
Assertion
Ref Expression
opropabco  |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
Distinct variable groups:    x, A, y    y, B    y, C    x, M, y    x, R, y    x, S, y
Allowed substitution hints:    B( x)    C( x)    F( x, y)    G( x, y)

Proof of Theorem opropabco
StepHypRef Expression
1 opropabco.1 . . 3  |-  ( x  e.  A  ->  B  e.  R )
2 opropabco.2 . . 3  |-  ( x  e.  A  ->  C  e.  S )
3 opelxpi 4721 . . 3  |-  ( ( B  e.  R  /\  C  e.  S )  -> 
<. B ,  C >.  e.  ( R  X.  S
) )
41, 2, 3syl2anc 642 . 2  |-  ( x  e.  A  ->  <. B ,  C >.  e.  ( R  X.  S ) )
5 opropabco.3 . 2  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  <. B ,  C >. ) }
6 opropabco.4 . . 3  |-  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( B M C ) ) }
7 df-ov 5861 . . . . . 6  |-  ( B M C )  =  ( M `  <. B ,  C >. )
87eqeq2i 2293 . . . . 5  |-  ( y  =  ( B M C )  <->  y  =  ( M `  <. B ,  C >. ) )
98anbi2i 675 . . . 4  |-  ( ( x  e.  A  /\  y  =  ( B M C ) )  <->  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) )
109opabbii 4083 . . 3  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( B M C ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) }
116, 10eqtri 2303 . 2  |-  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) }
124, 5, 11fnopabco 26388 1  |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   {copab 4076    X. cxp 4687    o. ccom 4693    Fn wfn 5250   ` cfv 5255  (class class class)co 5858
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
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-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-ov 5861
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