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Theorem istsr2 17427
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 17426 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
3 qfto 5659 . . 3 ((𝑋 × 𝑋) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥))
43anbi2i 603 . 2 ((𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)) ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
52, 4bitri 264 1 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wo 828   = wceq 1631  wcel 2145  wral 3061  cun 3722  wss 3724   class class class wbr 4787   × cxp 5248  ccnv 5249  dom cdm 5250  PosetRelcps 17407   TosetRel ctsr 17408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258  df-dm 5260  df-tsr 17410
This theorem is referenced by:  tsrlin  17428  tsrss  17432
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