Theorem caovcl 6082

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 1368	. 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 6080	. 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 1365   e. wcel 2167  (class class class)co 5925

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  ecopovtrn  6700  ecopovtrng  6703  genpelvl  7596  genpelvu  7597  genpml  7601  genpmu  7602  genprndl  7605  genprndu  7606  genpassl  7608  genpassu  7609  genpassg  7610  expcllem  10659