Theorem caovcld 5685

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 5684	. 2 |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )
6	1, 2, 3, 5	syl12anc 1168	1 |- ( ph -> ( A F B ) e. E )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1434  (class class class)co 5543

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546

This theorem is referenced by:  caovdir2d  5708  caov4d  5716  caovdilemd  5723  caovlem2d  5724  grprinvd  5727  ecopovtrn  6269  ecopovtrng  6272  ordpipqqs  6626  ltanqg  6652  ltmnqg  6653  recexprlem1ssu  6886  mulgt0sr  7016  mulextsr1lem  7018  axmulass  7101  frec2uzrdg  9491  frecuzrdgsuc  9496  frecuzrdgsuctlem  9505  iseqovex  9529  iseqval  9530  iseqvalt  9532  iseqfclt  9536  iseqp1  9538  iseqp1t  9539  iseqdistr  9567  climcn2  10286