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Theorem caovclg 5797
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovclg.1  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
Assertion
Ref Expression
caovclg  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
Distinct variable groups:    x, y, A   
y, B    x, C, y    x, D, y    x, E, y    ph, x, y   
x, F, y
Allowed substitution hint:    B( x)

Proof of Theorem caovclg
StepHypRef Expression
1 caovclg.1 . . 3  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
21ralrimivva 2455 . 2  |-  ( ph  ->  A. x  e.  C  A. y  e.  D  ( x F y )  e.  E )
3 oveq1 5659 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43eleq1d 2156 . . 3  |-  ( x  =  A  ->  (
( x F y )  e.  E  <->  ( A F y )  e.  E ) )
5 oveq2 5660 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
65eleq1d 2156 . . 3  |-  ( y  =  B  ->  (
( A F y )  e.  E  <->  ( A F B )  e.  E
) )
74, 6rspc2v 2734 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ( x F y )  e.  E  ->  ( A F B )  e.  E ) )
82, 7mpan9 275 1  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359  (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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
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-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  caovcld  5798  caovcl  5799  caovlem2d  5837  grprinvd  5840  frec2uzrdg  9816  frecuzrdgsuc  9821  iseqovex  9870  iseqval  9871  iseqvalt  9873  seq3val  9874  iseqfclt  9879  seqf  9880  iseqcaopr  9912
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