ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  coss2 Unicode version

Theorem coss2 4744
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 4009 . . . . 5  |-  ( A 
C_  B  ->  (
x A y  ->  x B y ) )
32anim1d 334 . . . 4  |-  ( A 
C_  B  ->  (
( x A y  /\  y C z )  ->  ( x B y  /\  y C z ) ) )
43eximdv 1860 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x A y  /\  y C z )  ->  E. y
( x B y  /\  y C z ) ) )
54ssopab2dv 4240 . 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 4597 . 2  |-  ( C  o.  A )  =  { <. x ,  z
>.  |  E. y
( x A y  /\  y C z ) }
7 df-co 4597 . 2  |-  ( C  o.  B )  =  { <. x ,  z
>.  |  E. y
( x B y  /\  y C z ) }
85, 6, 73sstr4g 3171 1  |-  ( A 
C_  B  ->  ( C  o.  A )  C_  ( C  o.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1472    C_ wss 3102   class class class wbr 3967   {copab 4026    o. ccom 4592
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115  df-br 3968  df-opab 4028  df-co 4597
This theorem is referenced by:  coeq2  4746  funss  5191  tposss  6195  dftpos4  6212
  Copyright terms: Public domain W3C validator