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Theorem caovcld 5917
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 5916 . 2  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
61, 2, 3, 5syl12anc 1214 1  |-  ( ph  ->  ( A F B )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480  (class class class)co 5767
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770
This theorem is referenced by:  caovdir2d  5940  caov4d  5948  caovdilemd  5955  caovlem2d  5956  grprinvd  5959  ecopovtrn  6519  ecopovtrng  6522  ordpipqqs  7175  ltanqg  7201  ltmnqg  7202  recexprlem1ssu  7435  mulgt0sr  7579  mulextsr1lem  7581  axmulass  7674  frec2uzrdg  10175  frecuzrdgsuc  10180  frecuzrdgsuctlem  10189  iseqovex  10222  seq3val  10224  seqf  10227  seq3p1  10228  seqp1cd  10232  seq3clss  10233  seq3distr  10279  climcn2  11071
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