Theorem caovcl 5879

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 1318	. 2 |- T.
2		caovcl.1	. . . 4 |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S )
3	2	adantl 273	. . 3 |- ( ( T. /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
4	3	caovclg 5877	. 2 |- ( ( T. /\ ( A e. S /\ B e. S ) ) -> ( A F B ) e. S )
5	1, 4	mpan 418	1 |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S )

Syntax hints:   -> wi 4   /\ wa 103   T. wtru 1315   e. wcel 1463  (class class class)co 5728

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-iota 5046  df-fv 5089  df-ov 5731

This theorem is referenced by:  ecopovtrn  6480  ecopovtrng  6483  genpelvl  7268  genpelvu  7269  genpml  7273  genpmu  7274  genprndl  7277  genprndu  7278  genpassl  7280  genpassu  7281  genpassg  7282  expcllem  10197