Theorem coss2 4874

Description: Subclass theorem for composition. (Contributed by set.mm contributors, 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 4681	. . . . 5 |- (A C_ B -> (xAy -> xBy))
3	2	anim1d 547	. . . 4 |- (A C_ B -> ((xAy /\ yCz) -> (xBy /\ yCz)))
4	3	eximdv 1622	. . 3 |- (A C_ B -> (E.y(xAy /\ yCz) -> E.y(xBy /\ yCz)))
5	4	ssopab2dv 4716	. 2 |- (A C_ B -> {<.x, z>. | E.y(xAy /\ yCz)} C_ {<.x, z>. | E.y(xBy /\ yCz)})
6		df-co 4727	. 2 |- (C o. A) = {<.x, z>. | E.y(xAy /\ yCz)}
7		df-co 4727	. 2 |- (C o. B) = {<.x, z>. | E.y(xBy /\ yCz)}
8	5, 6, 7	3sstr4g 3313	1 |- (A C_ B -> (C o. A) C_ (C o. B))

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   C_ wss 3258  {copab 4623   class class class wbr 4640   o. ccom 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727

This theorem is referenced by:  coeq2  4876  funss  5127