Theorem caovcl 6126

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 1377	. 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 6124	. 2 |- ( ( T. /\ ( A e. S /\ B e. S ) ) -> ( A F B ) e. S )
5	1, 4	mpan 424	1 |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   T. wtru 1374   e. wcel 2178  (class class class)co 5969

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2779  df-un 3179  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-br 4061  df-iota 5252  df-fv 5299  df-ov 5972

This theorem is referenced by:  ecopovtrn  6744  ecopovtrng  6747  genpelvl  7662  genpelvu  7663  genpml  7667  genpmu  7668  genprndl  7671  genprndu  7672  genpassl  7674  genpassu  7675  genpassg  7676  expcllem  10734