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Theorem cnvss 4680
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 3059 . . . 4  |-  ( A 
C_  B  ->  ( <. y ,  x >.  e.  A  ->  <. y ,  x >.  e.  B
) )
2 df-br 3898 . . . 4  |-  ( y A x  <->  <. y ,  x >.  e.  A
)
3 df-br 3898 . . . 4  |-  ( y B x  <->  <. y ,  x >.  e.  B
)
41, 2, 33imtr4g 204 . . 3  |-  ( A 
C_  B  ->  (
y A x  -> 
y B x ) )
54ssopab2dv 4168 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  y A x }  C_  {
<. x ,  y >.  |  y B x } )
6 df-cnv 4515 . 2  |-  `' A  =  { <. x ,  y
>.  |  y A x }
7 df-cnv 4515 . 2  |-  `' B  =  { <. x ,  y
>.  |  y B x }
85, 6, 73sstr4g 3108 1  |-  ( A 
C_  B  ->  `' A  C_  `' B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463    C_ wss 3039   <.cop 3498   class class class wbr 3897   {copab 3956   `'ccnv 4506
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-in 3045  df-ss 3052  df-br 3898  df-opab 3958  df-cnv 4515
This theorem is referenced by:  cnveq  4681  rnss  4737  relcnvtr  5026  funss  5110  funcnvuni  5160  funres11  5163  funcnvres  5164  foimacnv  5351  tposss  6109  structcnvcnv  11881
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