MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istsr2 Structured version   Visualization version   GIF version

Theorem istsr2 18533
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
istsr2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑋,𝑦

Proof of Theorem istsr2
StepHypRef Expression
1 istsr.1 . . 3 𝑋 = dom 𝑅
21istsr 18532 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
3 qfto 6119 . . 3 ((𝑋 × 𝑋) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥))
43anbi2i 624 . 2 ((𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)) ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
52, 4bitri 275 1 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3062  cun 3945  wss 3947   class class class wbr 5147   × cxp 5673  ccnv 5674  dom cdm 5675  PosetRelcps 18513   TosetRel ctsr 18514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-tsr 18516
This theorem is referenced by:  tsrlin  18534  tsrss  18538
  Copyright terms: Public domain W3C validator