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Theorem caovcl 5799
Description: Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovcl.1  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )
Assertion
Ref Expression
caovcl  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
Distinct variable groups:    x, y, A   
y, B    x, F, y    x, S, y
Allowed substitution hint:    B( x)

Proof of Theorem caovcl
StepHypRef Expression
1 tru 1293 . 2  |- T.
2 caovcl.1 . . . 4  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )
32adantl 271 . . 3  |-  ( ( T.  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
43caovclg 5797 . 2  |-  ( ( T.  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  e.  S )
51, 4mpan 415 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   T. wtru 1290    e. wcel 1438  (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:  ecopovtrn  6389  ecopovtrng  6392  genpelvl  7071  genpelvu  7072  genpml  7076  genpmu  7077  genprndl  7080  genprndu  7081  genpassl  7083  genpassu  7084  genpassg  7085  expcllem  9966
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