Theorem caovcl 6244

Description: Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Hypothesis

Ref	Expression
caovcl.1	|- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S )

Assertion

Ref	Expression
caovcl	|- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S )

Distinct variable groups:   x, y, A   y, B   x, F, y   x, S, y

Allowed substitution hint:   B( x)

Proof of Theorem caovcl

Step	Hyp	Ref	Expression
1		tru 1406	. 2 |- T.
2		caovcl.1	. . . 4 |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S )
3	2	adantl 277	. . 3 |- ( ( T. /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
4	3	caovclg 6242	. 2 |- ( ( T. /\ ( A e. S /\ B e. S ) ) -> ( A F B ) e. S )
5	1, 4	mpan 428	1 |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   T. wtru 1403   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:  ecopovtrn  6906  ecopovtrng  6909  genpelvl  7879  genpelvu  7880  genpml  7884  genpmu  7885  genprndl  7888  genprndu  7889  genpassl  7891  genpassu  7892  genpassg  7893  expcllem  10987