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Theorem opropabco 26425
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 4910 . . 3  |-  ( ( B  e.  R  /\  C  e.  S )  -> 
<. B ,  C >.  e.  ( R  X.  S
) )
41, 2, 3syl2anc 643 . 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 6084 . . . . . 6  |-  ( B M C )  =  ( M `  <. B ,  C >. )
87eqeq2i 2446 . . . . 5  |-  ( y  =  ( B M C )  <->  y  =  ( M `  <. B ,  C >. ) )
98anbi2i 676 . . . 4  |-  ( ( x  e.  A  /\  y  =  ( B M C ) )  <->  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) )
109opabbii 4272 . . 3  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( B M C ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) }
116, 10eqtri 2456 . 2  |-  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) }
124, 5, 11fnopabco 26424 1  |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817   {copab 4265    X. cxp 4876    o. ccom 4882    Fn wfn 5449   ` cfv 5454  (class class class)co 6081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084
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