Theorem coss1 4784

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 4048	. . . . 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 1880	. . 3 |- ( A C_ B -> ( E. y ( x C y /\ y A z ) -> E. y ( x C y /\ y B z ) ) )
5	4	ssopab2dv 4280	. 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 4637	. 2 |- ( A o. C ) = { <. x , z >. | E. y ( x C y /\ y A z ) }
7		df-co 4637	. 2 |- ( B o. C ) = { <. x , z >. | E. y ( x C y /\ y B z ) }
8	5, 6, 7	3sstr4g 3200	1 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1492   C_ wss 3131   class class class wbr 4005   {copab 4065   o. ccom 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3137  df-ss 3144  df-br 4006  df-opab 4067  df-co 4637

This theorem is referenced by:  coeq1  4786  funss  5237  tposss  6249