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Theorem opropabco 28764
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 4978 . . 3  |-  ( ( B  e.  R  /\  C  e.  S )  -> 
<. B ,  C >.  e.  ( R  X.  S
) )
41, 2, 3syl2anc 661 . 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 6202 . . . . . 6  |-  ( B M C )  =  ( M `  <. B ,  C >. )
87eqeq2i 2472 . . . . 5  |-  ( y  =  ( B M C )  <->  y  =  ( M `  <. B ,  C >. ) )
98anbi2i 694 . . . 4  |-  ( ( x  e.  A  /\  y  =  ( B M C ) )  <->  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) )
109opabbii 4463 . . 3  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( B M C ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) }
116, 10eqtri 2483 . 2  |-  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) }
124, 5, 11fnopabco 28763 1  |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3990   {copab 4456    X. cxp 4945    o. ccom 4951    Fn wfn 5520   ` cfv 5525  (class class class)co 6199
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 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6202
This theorem is referenced by: (None)
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