Theorem coss1 4873

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 4681	. . . . 5 |- (A C_ B -> (yAz -> yBz))
3	2	anim2d 548	. . . 4 |- (A C_ B -> ((xCy /\ yAz) -> (xCy /\ yBz)))
4	3	eximdv 1622	. . 3 |- (A C_ B -> (E.y(xCy /\ yAz) -> E.y(xCy /\ yBz)))
5	4	ssopab2dv 4716	. 2 |- (A C_ B -> {<.x, z>. | E.y(xCy /\ yAz)} C_ {<.x, z>. | E.y(xCy /\ yBz)})
6		df-co 4727	. 2 |- (A o. C) = {<.x, z>. | E.y(xCy /\ yAz)}
7		df-co 4727	. 2 |- (B o. C) = {<.x, z>. | E.y(xCy /\ yBz)}
8	5, 6, 7	3sstr4g 3313	1 |- (A C_ B -> (A o. C) C_ (B o. C))

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:  coeq1  4875  funss  5127