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Theorem coss2 5150
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 22 . . . . . 6  |-  ( A 
C_  B  ->  A  C_  B )
21ssbrd 4481 . . . . 5  |-  ( A 
C_  B  ->  (
x A y  ->  x B y ) )
32anim1d 564 . . . 4  |-  ( A 
C_  B  ->  (
( x A y  /\  y C z )  ->  ( x B y  /\  y C z ) ) )
43eximdv 1681 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x A y  /\  y C z )  ->  E. y
( x B y  /\  y C z ) ) )
54ssopab2dv 4769 . 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 5001 . 2  |-  ( C  o.  A )  =  { <. x ,  z
>.  |  E. y
( x A y  /\  y C z ) }
7 df-co 5001 . 2  |-  ( C  o.  B )  =  { <. x ,  z
>.  |  E. y
( x B y  /\  y C z ) }
85, 6, 73sstr4g 3538 1  |-  ( A 
C_  B  ->  ( C  o.  A )  C_  ( C  o.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1591    C_ wss 3469   class class class wbr 4440   {copab 4497    o. ccom 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-in 3476  df-ss 3483  df-br 4441  df-opab 4499  df-co 5001
This theorem is referenced by:  coeq2  5152  funss  5597  tposss  6946  dftpos4  6964  tsrdir  15714  mvdco  16259  ustex2sym  20447  ustex3sym  20448  ustund  20452  ustneism  20454  trust  20460  utop2nei  20481  neipcfilu  20527  fcoinver  27119  rtrclreclem.min  28531
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