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Theorem coss2 4840
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.
StepHypRef Expression
1 id 19 . . . . . 6  |-  ( A 
C_  B  ->  A  C_  B )
21ssbrd 4064 . . . . 5  |-  ( A 
C_  B  ->  (
x A y  ->  x B y ) )
32anim1d 547 . . . 4  |-  ( A 
C_  B  ->  (
( x A y  /\  y C z )  ->  ( x B y  /\  y C z ) ) )
43eximdv 1608 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x A y  /\  y C z )  ->  E. y
( x B y  /\  y C z ) ) )
54ssopab2dv 4293 . 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 4698 . 2  |-  ( C  o.  A )  =  { <. x ,  z
>.  |  E. y
( x A y  /\  y C z ) }
7 df-co 4698 . 2  |-  ( C  o.  B )  =  { <. x ,  z
>.  |  E. y
( x B y  /\  y C z ) }
85, 6, 73sstr4g 3219 1  |-  ( A 
C_  B  ->  ( C  o.  A )  C_  ( C  o.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    C_ wss 3152   class class class wbr 4023   {copab 4076    o. ccom 4693
This theorem is referenced by:  coeq2  4842  funss  5273  tposss  6235  dftpos4  6253  tsrdir  14360  rtrclreclem.min  24044  mvdco  27388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-in 3159  df-ss 3166  df-br 4024  df-opab 4078  df-co 4698
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