Theorem coss2 4936

Description: Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)

Assertion

Ref	Expression
coss2	|- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )

Proof of Theorem coss2

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 4173	. . . . 5 |- ( A C_ B -> ( x A y -> x B y ) )
3	2	anim1d 336	. . . 4 |- ( A C_ B -> ( ( x A y /\ y C z ) -> ( x B y /\ y C z ) ) )
4	3	eximdv 1933	. . 3 |- ( A C_ B -> ( E. y ( x A y /\ y C z ) -> E. y ( x B y /\ y C z ) ) )
5	4	ssopab2dv 4421	. 2 |- ( A C_ B -> { <. x , z >. | E. y ( x A y /\ y C z ) } C_ { <. x , z >. | E. y ( x B y /\ y C z ) } )
6		df-co 4783	. 2 |- ( C o. A ) = { <. x , z >. | E. y ( x A y /\ y C z ) }
7		df-co 4783	. 2 |- ( C o. B ) = { <. x , z >. | E. y ( x B y /\ y C z ) }
8	5, 6, 7	3sstr4g 3291	1 |- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   E.wex 1545   C_ wss 3220   class class class wbr 4130   {copab 4191   o. ccom 4778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-br 4131  df-opab 4193  df-co 4783

This theorem is used by:  coeq2  4938  funss  5396  tposss  6517  dftpos4  6534