Theorem trrelressn 38988

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38853) 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 38856	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
2		relres 5970	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dftrrel3 38983	. 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 712	1 ⊢ TrRel (𝑅 ↾ {𝐴})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540  {csn 4567   class class class wbr 5085   ↾ cres 5633  Rel wrel 5636   TrRel wtrrel 38519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-trrel 38979

This theorem is referenced by: (None)