Theorem coss2 4703

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 3979	. . . . 5 |- ( A C_ B -> ( x A y -> x B y ) )
3	2	anim1d 334	. . . 4 |- ( A C_ B -> ( ( x A y /\ y C z ) -> ( x B y /\ y C z ) ) )
4	3	eximdv 1853	. . 3 |- ( A C_ B -> ( E. y ( x A y /\ y C z ) -> E. y ( x B y /\ y C z ) ) )
5	4	ssopab2dv 4208	. 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 4556	. 2 |- ( C o. A ) = { <. x , z >. | E. y ( x A y /\ y C z ) }
7		df-co 4556	. 2 |- ( C o. B ) = { <. x , z >. | E. y ( x B y /\ y C z ) }
8	5, 6, 7	3sstr4g 3145	1 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1469   C_ wss 3076   class class class wbr 3937   {copab 3996   o. ccom 4551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3082  df-ss 3089  df-br 3938  df-opab 3998  df-co 4556

This theorem is referenced by:  coeq2  4705  funss  5150  tposss  6151  dftpos4  6168