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Theorem coss1 4800
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 4061 . . . . 5  |-  ( A 
C_  B  ->  (
y A z  -> 
y B z ) )
32anim2d 337 . . . 4  |-  ( A 
C_  B  ->  (
( x C y  /\  y A z )  ->  ( x C y  /\  y B z ) ) )
43eximdv 1891 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x C y  /\  y A z )  ->  E. y
( x C y  /\  y B z ) ) )
54ssopab2dv 4296 . 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 4653 . 2  |-  ( A  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y A z ) }
7 df-co 4653 . 2  |-  ( B  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y B z ) }
85, 6, 73sstr4g 3213 1  |-  ( A 
C_  B  ->  ( A  o.  C )  C_  ( B  o.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1503    C_ wss 3144   class class class wbr 4018   {copab 4078    o. ccom 4648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-in 3150  df-ss 3157  df-br 4019  df-opab 4080  df-co 4653
This theorem is referenced by:  coeq1  4802  funss  5254  tposss  6270
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