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Theorem coss1 4689
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 19 . . . . . 6  |-  ( A 
C_  B  ->  A  C_  B )
21ssbrd 3966 . . . . 5  |-  ( A 
C_  B  ->  (
y A z  -> 
y B z ) )
32anim2d 335 . . . 4  |-  ( A 
C_  B  ->  (
( x C y  /\  y A z )  ->  ( x C y  /\  y B z ) ) )
43eximdv 1852 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x C y  /\  y A z )  ->  E. y
( x C y  /\  y B z ) ) )
54ssopab2dv 4195 . 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 4543 . 2  |-  ( A  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y A z ) }
7 df-co 4543 . 2  |-  ( B  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y B z ) }
85, 6, 73sstr4g 3135 1  |-  ( A 
C_  B  ->  ( A  o.  C )  C_  ( B  o.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1468    C_ wss 3066   class class class wbr 3924   {copab 3983    o. ccom 4538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-br 3925  df-opab 3985  df-co 4543
This theorem is referenced by:  coeq1  4691  funss  5137  tposss  6136
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