Theorem caovcld 6074

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 6073	. 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 2164  (class class class)co 5919

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922

This theorem is referenced by:  caovdir2d  6097  caov4d  6105  caovdilemd  6112  caovlem2d  6113  ecopovtrn  6688  ecopovtrng  6691  ordpipqqs  7436  ltanqg  7462  ltmnqg  7463  recexprlem1ssu  7696  mulgt0sr  7840  mulextsr1lem  7842  axmulass  7935  frec2uzrdg  10483  frecuzrdgsuc  10488  frecuzrdgsuctlem  10497  iseqovex  10532  seq3val  10534  seqf  10538  seq3p1  10539  seqp1cd  10544  seq3clss  10545  seq3distr  10606  climcn2  11455  qusaddvallemg  12919  grpinva  12972