Theorem caovcld 6081

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 6080	. 2 |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )
6	1, 2, 3, 5	syl12anc 1247	1 |- ( ph -> ( A F B ) e. E )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167  (class class class)co 5925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  caovdir2d  6104  caov4d  6112  caovdilemd  6119  caovlem2d  6120  ecopovtrn  6700  ecopovtrng  6703  ordpipqqs  7458  ltanqg  7484  ltmnqg  7485  recexprlem1ssu  7718  mulgt0sr  7862  mulextsr1lem  7864  axmulass  7957  frec2uzrdg  10518  frecuzrdgsuc  10523  frecuzrdgsuctlem  10532  iseqovex  10567  seq3val  10569  seqf  10573  seq3p1  10574  seqp1cd  10579  seq3clss  10580  seq3distr  10641  climcn2  11491  qusaddvallemg  13035  grpinva  13088