Theorem coss1 4851

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 4102	. . . . 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 1904	. . 3 |- ( A C_ B -> ( E. y ( x C y /\ y A z ) -> E. y ( x C y /\ y B z ) ) )
5	4	ssopab2dv 4343	. 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 4702	. 2 |- ( A o. C ) = { <. x , z >. | E. y ( x C y /\ y A z ) }
7		df-co 4702	. 2 |- ( B o. C ) = { <. x , z >. | E. y ( x C y /\ y B z ) }
8	5, 6, 7	3sstr4g 3244	1 |- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1516   C_ wss 3174   class class class wbr 4059   {copab 4120   o. ccom 4697

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187  df-br 4060  df-opab 4122  df-co 4702

This theorem is referenced by:  coeq1  4853  funss  5309  tposss  6355