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Theorem istsr 15974
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 5213 . . . . 5  |-  ( r  =  R  ->  dom  r  =  dom  R )
2 istsr.1 . . . . 5  |-  X  =  dom  R
31, 2syl6eqr 2516 . . . 4  |-  ( r  =  R  ->  dom  r  =  X )
43sqxpeqd 5034 . . 3  |-  ( r  =  R  ->  ( dom  r  X.  dom  r
)  =  ( X  X.  X ) )
5 id 22 . . . 4  |-  ( r  =  R  ->  r  =  R )
6 cnveq 5186 . . . 4  |-  ( r  =  R  ->  `' r  =  `' R
)
75, 6uneq12d 3655 . . 3  |-  ( r  =  R  ->  (
r  u.  `' r )  =  ( R  u.  `' R ) )
84, 7sseq12d 3528 . 2  |-  ( r  =  R  ->  (
( dom  r  X.  dom  r )  C_  (
r  u.  `' r )  <->  ( X  X.  X )  C_  ( R  u.  `' R
) ) )
9 df-tsr 15958 . 2  |-  TosetRel  =  {
r  e.  PosetRel  |  ( dom  r  X.  dom  r )  C_  (
r  u.  `' r ) }
108, 9elrab2 3259 1  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( X  X.  X
)  C_  ( R  u.  `' R ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    u. cun 3469    C_ wss 3471    X. cxp 5006   `'ccnv 5007   dom cdm 5008   PosetRelcps 15955    TosetRel ctsr 15956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-cnv 5016  df-dm 5018  df-tsr 15958
This theorem is referenced by:  istsr2  15975  tsrlemax  15977  tsrps  15978  cnvtsr  15979  letsr  15984  tsrdir  15995
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