Theorem coss2 4744

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 4009	. . . . 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 1860	. . 3 |- ( A C_ B -> ( E. y ( x A y /\ y C z ) -> E. y ( x B y /\ y C z ) ) )
5	4	ssopab2dv 4240	. 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 4597	. 2 |- ( C o. A ) = { <. x , z >. | E. y ( x A y /\ y C z ) }
7		df-co 4597	. 2 |- ( C o. B ) = { <. x , z >. | E. y ( x B y /\ y C z ) }
8	5, 6, 7	3sstr4g 3171	1 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1472   C_ wss 3102   class class class wbr 3967   {copab 4026   o. ccom 4592

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115  df-br 3968  df-opab 4028  df-co 4597

This theorem is referenced by:  coeq2  4746  funss  5191  tposss  6195  dftpos4  6212