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Theorem istsr 16173
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 5026 . . . . 5  |-  ( r  =  R  ->  dom  r  =  dom  R )
2 istsr.1 . . . . 5  |-  X  =  dom  R
31, 2syl6eqr 2463 . . . 4  |-  ( r  =  R  ->  dom  r  =  X )
43sqxpeqd 4851 . . 3  |-  ( r  =  R  ->  ( dom  r  X.  dom  r
)  =  ( X  X.  X ) )
5 id 23 . . . 4  |-  ( r  =  R  ->  r  =  R )
6 cnveq 4999 . . . 4  |-  ( r  =  R  ->  `' r  =  `' R
)
75, 6uneq12d 3600 . . 3  |-  ( r  =  R  ->  (
r  u.  `' r )  =  ( R  u.  `' R ) )
84, 7sseq12d 3473 . 2  |-  ( r  =  R  ->  (
( dom  r  X.  dom  r )  C_  (
r  u.  `' r )  <->  ( X  X.  X )  C_  ( R  u.  `' R
) ) )
9 df-tsr 16157 . 2  |-  TosetRel  =  {
r  e.  PosetRel  |  ( dom  r  X.  dom  r )  C_  (
r  u.  `' r ) }
108, 9elrab2 3211 1  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( X  X.  X
)  C_  ( R  u.  `' R ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844    u. cun 3414    C_ wss 3416    X. cxp 4823   `'ccnv 4824   dom cdm 4825   PosetRelcps 16154    TosetRel ctsr 16155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-cnv 4833  df-dm 4835  df-tsr 16157
This theorem is referenced by:  istsr2  16174  tsrlemax  16176  tsrps  16177  cnvtsr  16178  letsr  16183  tsrdir  16194
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