Theorem coss2 4510

Description: Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)

Assertion

Ref	Expression
coss2	|- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )

Proof of Theorem coss2

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 |- ( A C_ B -> A C_ B )
2	1	ssbrd 3826	. . . . 5 |- ( A C_ B -> ( x A y -> x B y ) )
3	2	anim1d 329	. . . 4 |- ( A C_ B -> ( ( x A y /\ y C z ) -> ( x B y /\ y C z ) ) )
4	3	eximdv 1801	. . 3 |- ( A C_ B -> ( E. y ( x A y /\ y C z ) -> E. y ( x B y /\ y C z ) ) )
5	4	ssopab2dv 4033	. 2 |- ( A C_ B -> { <. x , z >. | E. y ( x A y /\ y C z ) } C_ { <. x , z >. | E. y ( x B y /\ y C z ) } )
6		df-co 4372	. 2 |- ( C o. A ) = { <. x , z >. | E. y ( x A y /\ y C z ) }
7		df-co 4372	. 2 |- ( C o. B ) = { <. x , z >. | E. y ( x B y /\ y C z ) }
8	5, 6, 7	3sstr4g 3040	1 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )

Syntax hints:   -> wi 4   /\ wa 102   E.wex 1421   C_ wss 2973   class class class wbr 3785   {copab 3838   o. ccom 4367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-br 3786  df-opab 3840  df-co 4372

This theorem is referenced by:  coeq2  4512  funss  4940  tposss  5884  dftpos4  5901