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Theorem cnvss 4536
Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
cnvss  |-  ( A 
C_  B  ->  `' A  C_  `' B )

Proof of Theorem cnvss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 2994 . . . 4  |-  ( A 
C_  B  ->  ( <. y ,  x >.  e.  A  ->  <. y ,  x >.  e.  B
) )
2 df-br 3794 . . . 4  |-  ( y A x  <->  <. y ,  x >.  e.  A
)
3 df-br 3794 . . . 4  |-  ( y B x  <->  <. y ,  x >.  e.  B
)
41, 2, 33imtr4g 203 . . 3  |-  ( A 
C_  B  ->  (
y A x  -> 
y B x ) )
54ssopab2dv 4041 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  y A x }  C_  {
<. x ,  y >.  |  y B x } )
6 df-cnv 4379 . 2  |-  `' A  =  { <. x ,  y
>.  |  y A x }
7 df-cnv 4379 . 2  |-  `' B  =  { <. x ,  y
>.  |  y B x }
85, 6, 73sstr4g 3041 1  |-  ( A 
C_  B  ->  `' A  C_  `' B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434    C_ wss 2974   <.cop 3409   class class class wbr 3793   {copab 3846   `'ccnv 4370
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-br 3794  df-opab 3848  df-cnv 4379
This theorem is referenced by:  cnveq  4537  rnss  4592  relcnvtr  4870  funss  4950  funcnvuni  4999  funres11  5002  funcnvres  5003  foimacnv  5175  tposss  5895
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