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.

Step	Hyp	Ref	Expression
1		trressn 37973	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
2		relres 6005	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dftrrel3 38106	. 2 ⊢ ( TrRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧) ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	mpbir2an 709	1 ⊢ TrRel (𝑅 ↾ {𝐴})

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)