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Theorem ovconst2 5677
Description: The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
Hypothesis
Ref Expression
oprvalconst2.1  |-  C  e. 
_V
Assertion
Ref Expression
ovconst2  |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( R ( ( A  X.  B )  X.  { C }
) S )  =  C )

Proof of Theorem ovconst2
StepHypRef Expression
1 df-ov 5540 . 2  |-  ( R ( ( A  X.  B )  X.  { C } ) S )  =  ( ( ( A  X.  B )  X.  { C }
) `  <. R ,  S >. )
2 opelxpi 4396 . . 3  |-  ( ( R  e.  A  /\  S  e.  B )  -> 
<. R ,  S >.  e.  ( A  X.  B
) )
3 oprvalconst2.1 . . . 4  |-  C  e. 
_V
43fvconst2 5403 . . 3  |-  ( <. R ,  S >.  e.  ( A  X.  B
)  ->  ( (
( A  X.  B
)  X.  { C } ) `  <. R ,  S >. )  =  C )
52, 4syl 14 . 2  |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( ( ( A  X.  B )  X. 
{ C } ) `
 <. R ,  S >. )  =  C )
61, 5syl5eq 2126 1  |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( R ( ( A  X.  B )  X.  { C }
) S )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   {csn 3400   <.cop 3403    X. cxp 4363   ` cfv 4926  (class class class)co 5537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fv 4934  df-ov 5540
This theorem is referenced by: (None)
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