Theorem coss1 4885

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 4131	. . . . 5 |- ( A C_ B -> ( y A z -> y B z ) )
3	2	anim2d 337	. . . 4 |- ( A C_ B -> ( ( x C y /\ y A z ) -> ( x C y /\ y B z ) ) )
4	3	eximdv 1928	. . 3 |- ( A C_ B -> ( E. y ( x C y /\ y A z ) -> E. y ( x C y /\ y B z ) ) )
5	4	ssopab2dv 4373	. 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 4734	. 2 |- ( A o. C ) = { <. x , z >. | E. y ( x C y /\ y A z ) }
7		df-co 4734	. 2 |- ( B o. C ) = { <. x , z >. | E. y ( x C y /\ y B z ) }
8	5, 6, 7	3sstr4g 3270	1 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1540   C_ wss 3200   class class class wbr 4088   {copab 4149   o. ccom 4729

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-br 4089  df-opab 4151  df-co 4734

This theorem is referenced by:  coeq1  4887  funss  5345  tposss  6411