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Theorem istsr 15399
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 5052 . . . . 5  |-  ( r  =  R  ->  dom  r  =  dom  R )
2 istsr.1 . . . . 5  |-  X  =  dom  R
31, 2syl6eqr 2493 . . . 4  |-  ( r  =  R  ->  dom  r  =  X )
43, 3xpeq12d 4877 . . 3  |-  ( r  =  R  ->  ( dom  r  X.  dom  r
)  =  ( X  X.  X ) )
5 id 22 . . . 4  |-  ( r  =  R  ->  r  =  R )
6 cnveq 5025 . . . 4  |-  ( r  =  R  ->  `' r  =  `' R
)
75, 6uneq12d 3523 . . 3  |-  ( r  =  R  ->  (
r  u.  `' r )  =  ( R  u.  `' R ) )
84, 7sseq12d 3397 . 2  |-  ( r  =  R  ->  (
( dom  r  X.  dom  r )  C_  (
r  u.  `' r )  <->  ( X  X.  X )  C_  ( R  u.  `' R
) ) )
9 df-tsr 15383 . 2  |-  TosetRel  =  {
r  e.  PosetRel  |  ( dom  r  X.  dom  r )  C_  (
r  u.  `' r ) }
108, 9elrab2 3131 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 1369    e. wcel 1756    u. cun 3338    C_ wss 3340    X. cxp 4850   `'ccnv 4851   dom cdm 4852   PosetRelcps 15380    TosetRel ctsr 15381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-xp 4858  df-cnv 4860  df-dm 4862  df-tsr 15383
This theorem is referenced by:  istsr2  15400  tsrlemax  15402  tsrps  15403  cnvtsr  15404  letsr  15409  tsrdir  15420
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