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Theorem coss1 5158
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 4488 . . . . 5  |-  ( A 
C_  B  ->  (
y A z  -> 
y B z ) )
32anim2d 565 . . . 4  |-  ( A 
C_  B  ->  (
( x C y  /\  y A z )  ->  ( x C y  /\  y B z ) ) )
43eximdv 1686 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x C y  /\  y A z )  ->  E. y
( x C y  /\  y B z ) ) )
54ssopab2dv 4776 . 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 5008 . 2  |-  ( A  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y A z ) }
7 df-co 5008 . 2  |-  ( B  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y B z ) }
85, 6, 73sstr4g 3545 1  |-  ( A 
C_  B  ->  ( A  o.  C )  C_  ( B  o.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1596    C_ wss 3476   class class class wbr 4447   {copab 4504    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-in 3483  df-ss 3490  df-br 4448  df-opab 4506  df-co 5008
This theorem is referenced by:  coeq1  5160  funss  5606  tposss  6956  tsrdir  15725  ustex2sym  20482  ustex3sym  20483  ustund  20487  ustneism  20489  trust  20495  utop2nei  20516  neipcfilu  20562  rtrclreclem.min  28573
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