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Theorem istsr 14326
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 4879 . . . . 5  |-  ( r  =  R  ->  dom  r  =  dom  R )
2 istsr.1 . . . . 5  |-  X  =  dom  R
31, 2syl6eqr 2333 . . . 4  |-  ( r  =  R  ->  dom  r  =  X )
43, 3xpeq12d 4714 . . 3  |-  ( r  =  R  ->  ( dom  r  X.  dom  r
)  =  ( X  X.  X ) )
5 id 19 . . . 4  |-  ( r  =  R  ->  r  =  R )
6 cnveq 4855 . . . 4  |-  ( r  =  R  ->  `' r  =  `' R
)
75, 6uneq12d 3330 . . 3  |-  ( r  =  R  ->  (
r  u.  `' r )  =  ( R  u.  `' R ) )
84, 7sseq12d 3207 . 2  |-  ( r  =  R  ->  (
( dom  r  X.  dom  r )  C_  (
r  u.  `' r )  <->  ( X  X.  X )  C_  ( R  u.  `' R
) ) )
9 df-tsr 14307 . 2  |-  TosetRel  =  {
r  e.  PosetRel  |  ( dom  r  X.  dom  r )  C_  (
r  u.  `' r ) }
108, 9elrab2 2925 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 1623    e. wcel 1684    u. cun 3150    C_ wss 3152    X. cxp 4687   `'ccnv 4688   dom cdm 4689   PosetRelcps 14301    TosetRel ctsr 14302
This theorem is referenced by:  istsr2  14327  tsrlemax  14329  tsrps  14330  cnvtsr  14331  letsr  14349  tsrdir  14360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-tsr 14307
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