Theorem coss1 4759

Description: Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)

Assertion

Ref	Expression
coss1	|- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )

Proof of Theorem coss1

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 4025	. . . . 5 |- ( A C_ B -> ( y A z -> y B z ) )
3	2	anim2d 335	. . . 4 |- ( A C_ B -> ( ( x C y /\ y A z ) -> ( x C y /\ y B z ) ) )
4	3	eximdv 1868	. . 3 |- ( A C_ B -> ( E. y ( x C y /\ y A z ) -> E. y ( x C y /\ y B z ) ) )
5	4	ssopab2dv 4256	. 2 |- ( A C_ B -> { <. x , z >. | E. y ( x C y /\ y A z ) } C_ { <. x , z >. | E. y ( x C y /\ y B z ) } )
6		df-co 4613	. 2 |- ( A o. C ) = { <. x , z >. | E. y ( x C y /\ y A z ) }
7		df-co 4613	. 2 |- ( B o. C ) = { <. x , z >. | E. y ( x C y /\ y B z ) }
8	5, 6, 7	3sstr4g 3185	1 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   C_ wss 3116   class class class wbr 3982   {copab 4042   o. ccom 4608

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044  df-co 4613

This theorem is referenced by:  coeq1  4761  funss  5207  tposss  6214