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Theorem caovcl 6244
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 1331 . 2  |-  T.
2 caovcl.1 . . . 4  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )
32adantl 454 . . 3  |-  ( (  T.  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
43caovclg 6242 . 2  |-  ( (  T.  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  e.  S )
51, 4mpan 653 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    T. wtru 1326    e. wcel 1726  (class class class)co 6084
This theorem is referenced by:  ecopovtrn  7010  eceqoveq  7012  genpss  8886  genpnnp  8887  genpass  8891  expcllem  11397  txlly  17673  txnlly  17674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087
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