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Theorem istsr 14342
Description: The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Hypothesis
Ref Expression
istsr.1  |-  X  =  dom  R
Assertion
Ref Expression
istsr  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( X  X.  X
)  C_  ( R  u.  `' R ) ) )

Proof of Theorem istsr
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 dmeq 4895 . . . . 5  |-  ( r  =  R  ->  dom  r  =  dom  R )
2 istsr.1 . . . . 5  |-  X  =  dom  R
31, 2syl6eqr 2346 . . . 4  |-  ( r  =  R  ->  dom  r  =  X )
43, 3xpeq12d 4730 . . 3  |-  ( r  =  R  ->  ( dom  r  X.  dom  r
)  =  ( X  X.  X ) )
5 id 19 . . . 4  |-  ( r  =  R  ->  r  =  R )
6 cnveq 4871 . . . 4  |-  ( r  =  R  ->  `' r  =  `' R
)
75, 6uneq12d 3343 . . 3  |-  ( r  =  R  ->  (
r  u.  `' r )  =  ( R  u.  `' R ) )
84, 7sseq12d 3220 . 2  |-  ( r  =  R  ->  (
( dom  r  X.  dom  r )  C_  (
r  u.  `' r )  <->  ( X  X.  X )  C_  ( R  u.  `' R
) ) )
9 df-tsr 14323 . 2  |-  TosetRel  =  {
r  e.  PosetRel  |  ( dom  r  X.  dom  r )  C_  (
r  u.  `' r ) }
108, 9elrab2 2938 1  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( X  X.  X
)  C_  ( R  u.  `' R ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165    X. cxp 4703   `'ccnv 4704   dom cdm 4705   PosetRelcps 14317    TosetRel ctsr 14318
This theorem is referenced by:  istsr2  14343  tsrlemax  14345  tsrps  14346  cnvtsr  14347  letsr  14365  tsrdir  14376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-tsr 14323
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