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Theorem trrelressn 38111
Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37970) is transitive. (Contributed by Peter Mazsa, 17-Jun-2024.)
Assertion
Ref Expression
trrelressn TrRel (𝑅 ↾ {𝐴})

Proof of Theorem trrelressn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trressn 37973 . 2 𝑥𝑦𝑧((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
2 relres 6005 . 2 Rel (𝑅 ↾ {𝐴})
3 dftrrel3 38106 . 2 ( TrRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥𝑦𝑧((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧) ∧ Rel (𝑅 ↾ {𝐴})))
41, 2, 3mpbir2an 709 1 TrRel (𝑅 ↾ {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1531  {csn 4624   class class class wbr 5143  cres 5674  Rel wrel 5677   TrRel wtrrel 37720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-trrel 38102
This theorem is referenced by: (None)
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