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Theorem oprab2co 6760
Description: Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
Hypotheses
Ref Expression
oprab2co.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  R )
oprab2co.2  |-  ( ( x  e.  A  /\  y  e.  B )  ->  D  e.  S )
oprab2co.3  |-  F  =  ( x  e.  A ,  y  e.  B  |-> 
<. C ,  D >. )
oprab2co.4  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )
Assertion
Ref Expression
oprab2co  |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
Distinct variable groups:    x, y, A    x, B, y    x, M, y    x, R, y   
x, S, y
Allowed substitution hints:    C( x, y)    D( x, y)    F( x, y)    G( x, y)

Proof of Theorem oprab2co
StepHypRef Expression
1 oprab2co.1 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  R )
2 oprab2co.2 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  D  e.  S )
3 opelxpi 4971 . . 3  |-  ( ( C  e.  R  /\  D  e.  S )  -> 
<. C ,  D >.  e.  ( R  X.  S
) )
41, 2, 3syl2anc 661 . 2  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. C ,  D >.  e.  ( R  X.  S
) )
5 oprab2co.3 . 2  |-  F  =  ( x  e.  A ,  y  e.  B  |-> 
<. C ,  D >. )
6 oprab2co.4 . . 3  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )
7 df-ov 6195 . . . . 5  |-  ( C M D )  =  ( M `  <. C ,  D >. )
87a1i 11 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( C M D )  =  ( M `
 <. C ,  D >. ) )
98mpt2eq3ia 6252 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )  =  ( x  e.  A ,  y  e.  B  |->  ( M `  <. C ,  D >. ) )
106, 9eqtri 2480 . 2  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( M `  <. C ,  D >. )
)
114, 5, 10oprabco 6759 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 3983    X. cxp 4938    o. ccom 4944    Fn wfn 5513   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194
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 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680
This theorem is referenced by: (None)
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