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Theorem cnvss 5047
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 3344 . . . 4  |-  ( A 
C_  B  ->  ( <. y ,  x >.  e.  A  ->  <. y ,  x >.  e.  B
) )
2 df-br 4215 . . . 4  |-  ( y A x  <->  <. y ,  x >.  e.  A
)
3 df-br 4215 . . . 4  |-  ( y B x  <->  <. y ,  x >.  e.  B
)
41, 2, 33imtr4g 263 . . 3  |-  ( A 
C_  B  ->  (
y A x  -> 
y B x ) )
54ssopab2dv 4485 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  y A x }  C_  {
<. x ,  y >.  |  y B x } )
6 df-cnv 4888 . 2  |-  `' A  =  { <. x ,  y
>.  |  y A x }
7 df-cnv 4888 . 2  |-  `' B  =  { <. x ,  y
>.  |  y B x }
85, 6, 73sstr4g 3391 1  |-  ( A 
C_  B  ->  `' A  C_  `' B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726    C_ wss 3322   <.cop 3819   class class class wbr 4214   {copab 4267   `'ccnv 4879
This theorem is referenced by:  cnveq  5048  rnss  5100  relcnvtr  5391  funss  5474  funcnvuni  5520  funres11  5523  funcnvres  5524  foimacnv  5694  tposss  6482  vdwnnlem1  13365  structcnvcnv  13482  catcoppccl  14265  cnvps  14646  tsrdir  14685  gsumzres  15519  gsumzadd  15529  gsum2d  15548  dprdfadd  15580  tsmsres  18175  ustneism  18255  metustsymOLD  18593  metustsym  18594  metustOLD  18599  metust  18600  pi1xfrcnv  19084  tdeglem4  19985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-in 3329  df-ss 3336  df-br 4215  df-opab 4269  df-cnv 4888
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