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Theorem ovres 5640
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 4376 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  -> 
<. A ,  B >.  e.  ( C  X.  D
) )
2 fvres 5198 . . 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 5515 . 2  |-  ( A ( F  |`  ( C  X.  D ) ) B )  =  ( ( F  |`  ( C  X.  D ) ) `
 <. A ,  B >. )
5 df-ov 5515 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
63, 4, 53eqtr4g 2097 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 97    = wceq 1243    e. wcel 1393   <.cop 3378    X. cxp 4343    |` cres 4347   ` cfv 4902  (class class class)co 5512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-res 4357  df-iota 4867  df-fv 4910  df-ov 5515
This theorem is referenced by:  ovresd  5641  oprssov  5642  ofmresval  5723  elq  8557
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