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Theorem ovconst2 6430
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 6278 . 2  |-  ( R ( ( A  X.  B )  X.  { C } ) S )  =  ( ( ( A  X.  B )  X.  { C }
) `  <. R ,  S >. )
2 opelxpi 5023 . . 3  |-  ( ( R  e.  A  /\  S  e.  B )  -> 
<. R ,  S >.  e.  ( A  X.  B
) )
3 oprvalconst2.1 . . . 4  |-  C  e. 
_V
43fvconst2 6107 . . 3  |-  ( <. R ,  S >.  e.  ( A  X.  B
)  ->  ( (
( A  X.  B
)  X.  { C } ) `  <. R ,  S >. )  =  C )
52, 4syl 16 . 2  |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( ( ( A  X.  B )  X. 
{ C } ) `
 <. R ,  S >. )  =  C )
61, 5syl5eq 2513 1  |-  ( ( R  e.  A  /\  S  e.  B )  ->  ( R ( ( A  X.  B )  X.  { C }
) S )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   {csn 4020   <.cop 4026    X. cxp 4990   ` cfv 5579  (class class class)co 6275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278
This theorem is referenced by: (None)
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