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Theorem coss1 5000
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 4338 . . . . 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 1676 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x C y  /\  y A z )  ->  E. y
( x C y  /\  y B z ) ) )
54ssopab2dv 4622 . 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 4854 . 2  |-  ( A  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y A z ) }
7 df-co 4854 . 2  |-  ( B  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y B z ) }
85, 6, 73sstr4g 3402 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 1586    C_ wss 3333   class class class wbr 4297   {copab 4354    o. ccom 4849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-in 3340  df-ss 3347  df-br 4298  df-opab 4356  df-co 4854
This theorem is referenced by:  coeq1  5002  funss  5441  tposss  6751  tsrdir  15413  ustex2sym  19796  ustex3sym  19797  ustund  19801  ustneism  19803  trust  19809  utop2nei  19830  neipcfilu  19876  rtrclreclem.min  27354
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