Theorem caovcl 6828

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 1487	. 2 |- T.
2		caovcl.1	. . . 4 |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S )
3	2	adantl 482	. . 3 |- ( ( T. /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
4	3	caovclg 6826	. 2 |- ( ( T. /\ ( A e. S /\ B e. S ) ) -> ( A F B ) e. S )
5	1, 4	mpan 706	1 |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S )

Syntax hints:   -> wi 4   /\ wa 384   T. wtru 1484   e. wcel 1990  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  ecopovtrn  7850  eceqoveq  7853  genpss  9826  genpnnp  9827  genpass  9831  expcllem  12871  txlly  21439  txnlly  21440