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Theorem ovconst2 6083
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 5945 . 2  |-  ( R ( ( A  X.  B )  X.  { C } ) S )  =  ( ( ( A  X.  B )  X.  { C }
) `  <. R ,  S >. )
2 opelxpi 4800 . . 3  |-  ( ( R  e.  A  /\  S  e.  B )  -> 
<. R ,  S >.  e.  ( A  X.  B
) )
3 oprvalconst2.1 . . . 4  |-  C  e. 
_V
43fvconst2 5810 . . 3  |-  ( <. R ,  S >.  e.  ( A  X.  B
)  ->  ( (
( A  X.  B
)  X.  { C } ) `  <. R ,  S >. )  =  C )
52, 4syl 15 . 2  |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( ( ( A  X.  B )  X. 
{ C } ) `
 <. R ,  S >. )  =  C )
61, 5syl5eq 2402 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 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   {csn 3716   <.cop 3719    X. cxp 4766   ` cfv 5334  (class class class)co 5942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-ov 5945
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