MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovconst2 Unicode version

Theorem ovconst2 5999
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 5861 . 2  |-  ( R ( ( A  X.  B )  X.  { C } ) S )  =  ( ( ( A  X.  B )  X.  { C }
) `  <. R ,  S >. )
2 opelxpi 4721 . . 3  |-  ( ( R  e.  A  /\  S  e.  B )  -> 
<. R ,  S >.  e.  ( A  X.  B
) )
3 oprvalconst2.1 . . . 4  |-  C  e. 
_V
43fvconst2 5729 . . 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 2327 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 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643    X. cxp 4687   ` cfv 5255  (class class class)co 5858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861
  Copyright terms: Public domain W3C validator