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Theorem istsr2 18547
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 18546 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
3 qfto 6122 . . 3 ((𝑋 × 𝑋) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥))
43anbi2i 622 . 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 395  wo 844   = wceq 1540  wcel 2105  wral 3060  cun 3946  wss 3948   class class class wbr 5148   × cxp 5674  ccnv 5675  dom cdm 5676  PosetRelcps 18527   TosetRel ctsr 18528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-tsr 18530
This theorem is referenced by:  tsrlin  18548  tsrss  18552
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