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Theorem oprab2co 6425
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 4903 . . 3  |-  ( ( C  e.  R  /\  D  e.  S )  -> 
<. C ,  D >.  e.  ( R  X.  S
) )
41, 2, 3syl2anc 643 . 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 6077 . . . . 5  |-  ( C M D )  =  ( M `  <. C ,  D >. )
87a1i 11 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( C M D )  =  ( M `
 <. C ,  D >. ) )
98mpt2eq3ia 6132 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )  =  ( x  e.  A ,  y  e.  B  |->  ( M `  <. C ,  D >. ) )
106, 9eqtri 2456 . 2  |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( M `  <. C ,  D >. )
)
114, 5, 10oprabco 6424 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 3810    X. cxp 4869    o. ccom 4875    Fn wfn 5442   ` cfv 5447  (class class class)co 6074    e. cmpt2 6076
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 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343
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