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Theorem ovconst2 6188
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 6047 . 2  |-  ( R ( ( A  X.  B )  X.  { C } ) S )  =  ( ( ( A  X.  B )  X.  { C }
) `  <. R ,  S >. )
2 opelxpi 4873 . . 3  |-  ( ( R  e.  A  /\  S  e.  B )  -> 
<. R ,  S >.  e.  ( A  X.  B
) )
3 oprvalconst2.1 . . . 4  |-  C  e. 
_V
43fvconst2 5910 . . 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 2452 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2920   {csn 3778   <.cop 3781    X. cxp 4839   ` cfv 5417  (class class class)co 6044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047
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