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Theorem caovcld 5892
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovclg.1  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
caovcld.2  |-  ( ph  ->  A  e.  C )
caovcld.3  |-  ( ph  ->  B  e.  D )
Assertion
Ref Expression
caovcld  |-  ( ph  ->  ( 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 caovcld
StepHypRef Expression
1 id 19 . 2  |-  ( ph  ->  ph )
2 caovcld.2 . 2  |-  ( ph  ->  A  e.  C )
3 caovcld.3 . 2  |-  ( ph  ->  B  e.  D )
4 caovclg.1 . . 3  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
54caovclg 5891 . 2  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
61, 2, 3, 5syl12anc 1199 1  |-  ( ph  ->  ( A F B )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465  (class class class)co 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  caovdir2d  5915  caov4d  5923  caovdilemd  5930  caovlem2d  5931  grprinvd  5934  ecopovtrn  6494  ecopovtrng  6497  ordpipqqs  7150  ltanqg  7176  ltmnqg  7177  recexprlem1ssu  7410  mulgt0sr  7554  mulextsr1lem  7556  axmulass  7649  frec2uzrdg  10150  frecuzrdgsuc  10155  frecuzrdgsuctlem  10164  iseqovex  10197  seq3val  10199  seqf  10202  seq3p1  10203  seqp1cd  10207  seq3clss  10208  seq3distr  10254  climcn2  11046
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