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Theorem opropabco 30454
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 5020 . . 3  |-  ( ( B  e.  R  /\  C  e.  S )  -> 
<. B ,  C >.  e.  ( R  X.  S
) )
41, 2, 3syl2anc 659 . 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 6273 . . . . . 6  |-  ( B M C )  =  ( M `  <. B ,  C >. )
87eqeq2i 2472 . . . . 5  |-  ( y  =  ( B M C )  <->  y  =  ( M `  <. B ,  C >. ) )
98anbi2i 692 . . . 4  |-  ( ( x  e.  A  /\  y  =  ( B M C ) )  <->  ( x  e.  A  /\  y  =  ( M `  <. B ,  C >. ) ) )
109opabbii 4503 . . 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 30453 1  |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022   {copab 4496    X. cxp 4986    o. ccom 4992    Fn wfn 5565   ` cfv 5570  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273
This theorem is referenced by: (None)
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