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Theorem coss2 5029
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 20 . . . . . 6  |-  ( A 
C_  B  ->  A  C_  B )
21ssbrd 4253 . . . . 5  |-  ( A 
C_  B  ->  (
x A y  ->  x B y ) )
32anim1d 548 . . . 4  |-  ( A 
C_  B  ->  (
( x A y  /\  y C z )  ->  ( x B y  /\  y C z ) ) )
43eximdv 1632 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x A y  /\  y C z )  ->  E. y
( x B y  /\  y C z ) ) )
54ssopab2dv 4483 . 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 4887 . 2  |-  ( C  o.  A )  =  { <. x ,  z
>.  |  E. y
( x A y  /\  y C z ) }
7 df-co 4887 . 2  |-  ( C  o.  B )  =  { <. x ,  z
>.  |  E. y
( x B y  /\  y C z ) }
85, 6, 73sstr4g 3389 1  |-  ( A 
C_  B  ->  ( C  o.  A )  C_  ( C  o.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    C_ wss 3320   class class class wbr 4212   {copab 4265    o. ccom 4882
This theorem is referenced by:  coeq2  5031  funss  5472  tposss  6480  dftpos4  6498  tsrdir  14683  ustex2sym  18246  ustex3sym  18247  ustund  18251  ustneism  18253  trust  18259  utop2nei  18280  neipcfilu  18326  rtrclreclem.min  25147  mvdco  27365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-in 3327  df-ss 3334  df-br 4213  df-opab 4267  df-co 4887
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