Theorem coss2 4834

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 4087	. . . . 5 |- ( A C_ B -> ( x A y -> x B y ) )
3	2	anim1d 336	. . . 4 |- ( A C_ B -> ( ( x A y /\ y C z ) -> ( x B y /\ y C z ) ) )
4	3	eximdv 1903	. . 3 |- ( A C_ B -> ( E. y ( x A y /\ y C z ) -> E. y ( x B y /\ y C z ) ) )
5	4	ssopab2dv 4325	. 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 4684	. 2 |- ( C o. A ) = { <. x , z >. | E. y ( x A y /\ y C z ) }
7		df-co 4684	. 2 |- ( C o. B ) = { <. x , z >. | E. y ( x B y /\ y C z ) }
8	5, 6, 7	3sstr4g 3236	1 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1515   C_ wss 3166   class class class wbr 4044   {copab 4104   o. ccom 4679

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-br 4045  df-opab 4106  df-co 4684

This theorem is referenced by:  coeq2  4836  funss  5290  tposss  6332  dftpos4  6349