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Theorem caovcl 6269
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 1373 . 2  |- T.
2 caovcl.1 . . . 4  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )
32adantl 466 . . 3  |-  ( ( T.  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
43caovclg 6267 . 2  |-  ( ( T.  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  e.  S )
51, 4mpan 670 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   T. wtru 1370    e. wcel 1756  (class class class)co 6103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-iota 5393  df-fv 5438  df-ov 6106
This theorem is referenced by:  ecopovtrn  7215  eceqoveq  7217  genpss  9185  genpnnp  9186  genpass  9190  expcllem  11888  txlly  19221  txnlly  19222
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