Theorem caovcld 6243

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

Step	Hyp	Ref	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 )
5	4	caovclg 6242	. 2 |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )
6	1, 2, 3, 5	syl12anc 1276	1 |- ( ph -> ( A F B ) e. E )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  caovdir2d  6266  caov4d  6274  caovdilemd  6281  caovlem2d  6282  ecopovtrn  6906  ecopovtrng  6909  ordpipqqs  7742  ltanqg  7768  ltmnqg  7769  recexprlem1ssu  8002  mulgt0sr  8146  mulextsr1lem  8148  axmulass  8241  frec2uzrdg  10861  frecuzrdgsuc  10866  frecuzrdgsuctlem  10875  iseqovex  10910  seq3val  10912  seqf  10916  seq3p1  10917  seqp1cd  10922  seq3clss  10923  seq3distr  10984  climcn2  12094  qusaddvallemg  13707  grpinva  13759