Theorem coss1 4766

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 4032	. . . . 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 1873	. . 3 |- ( A C_ B -> ( E. y ( x C y /\ y A z ) -> E. y ( x C y /\ y B z ) ) )
5	4	ssopab2dv 4263	. 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 4620	. 2 |- ( A o. C ) = { <. x , z >. | E. y ( x C y /\ y A z ) }
7		df-co 4620	. 2 |- ( B o. C ) = { <. x , z >. | E. y ( x C y /\ y B z ) }
8	5, 6, 7	3sstr4g 3190	1 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1485   C_ wss 3121   class class class wbr 3989   {copab 4049   o. ccom 4615

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-br 3990  df-opab 4051  df-co 4620

This theorem is referenced by:  coeq1  4768  funss  5217  tposss  6225