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Theorem ovres 5784
Description: The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
Assertion
Ref Expression
ovres  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A ( F  |`  ( C  X.  D
) ) B )  =  ( A F B ) )

Proof of Theorem ovres
StepHypRef Expression
1 opelxpi 4469 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  -> 
<. A ,  B >.  e.  ( C  X.  D
) )
2 fvres 5329 . . 3  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  ->  ( ( F  |`  ( C  X.  D ) ) `  <. A ,  B >. )  =  ( F `  <. A ,  B >. ) )
31, 2syl 14 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F  |`  ( C  X.  D
) ) `  <. A ,  B >. )  =  ( F `  <. A ,  B >. ) )
4 df-ov 5655 . 2  |-  ( A ( F  |`  ( C  X.  D ) ) B )  =  ( ( F  |`  ( C  X.  D ) ) `
 <. A ,  B >. )
5 df-ov 5655 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
63, 4, 53eqtr4g 2145 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A ( F  |`  ( C  X.  D
) ) B )  =  ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   <.cop 3449    X. cxp 4436    |` cres 4440   ` cfv 5015  (class class class)co 5652
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-res 4450  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  ovresd  5785  oprssov  5786  ofmresval  5867  elq  9107
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