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Theorem coss2 4580
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 19 . . . . . 6  |-  ( A 
C_  B  ->  A  C_  B )
21ssbrd 3878 . . . . 5  |-  ( A 
C_  B  ->  (
x A y  ->  x B y ) )
32anim1d 329 . . . 4  |-  ( A 
C_  B  ->  (
( x A y  /\  y C z )  ->  ( x B y  /\  y C z ) ) )
43eximdv 1808 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x A y  /\  y C z )  ->  E. y
( x B y  /\  y C z ) ) )
54ssopab2dv 4096 . 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 4437 . 2  |-  ( C  o.  A )  =  { <. x ,  z
>.  |  E. y
( x A y  /\  y C z ) }
7 df-co 4437 . 2  |-  ( C  o.  B )  =  { <. x ,  z
>.  |  E. y
( x B y  /\  y C z ) }
85, 6, 73sstr4g 3065 1  |-  ( A 
C_  B  ->  ( C  o.  A )  C_  ( C  o.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1426    C_ wss 2997   class class class wbr 3837   {copab 3890    o. ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3003  df-ss 3010  df-br 3838  df-opab 3892  df-co 4437
This theorem is referenced by:  coeq2  4582  funss  5020  tposss  5993  dftpos4  6010
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