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Theorem coss1 5009
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.
StepHypRef Expression
1 id 22 . . . . . 6  |-  ( A 
C_  B  ->  A  C_  B )
21ssbrd 4465 . . . . 5  |-  ( A 
C_  B  ->  (
y A z  -> 
y B z ) )
32anim2d 567 . . . 4  |-  ( A 
C_  B  ->  (
( x C y  /\  y A z )  ->  ( x C y  /\  y B z ) ) )
43eximdv 1758 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x C y  /\  y A z )  ->  E. y
( x C y  /\  y B z ) ) )
54ssopab2dv 4749 . 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 4862 . 2  |-  ( A  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y A z ) }
7 df-co 4862 . 2  |-  ( B  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y B z ) }
85, 6, 73sstr4g 3505 1  |-  ( A 
C_  B  ->  ( A  o.  C )  C_  ( B  o.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   E.wex 1657    C_ wss 3436   class class class wbr 4423   {copab 4481    o. ccom 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-in 3443  df-ss 3450  df-br 4424  df-opab 4483  df-co 4862
This theorem is referenced by:  coeq1  5011  funss  5619  tposss  6986  rtrclreclem4  13125  tsrdir  16484  ustex2sym  21230  ustex3sym  21231  ustund  21235  ustneism  21237  trust  21243  utop2nei  21264  neipcfilu  21310  trclubgNEW  36196  trrelsuperrel2dg  36234  trclrelexplem  36274  cotrcltrcl  36288  cotrclrcl  36305  frege96d  36312  frege97d  36315  frege109d  36320  frege131d  36327
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