Theorem trrelressn 39005

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38870) 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 38873	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
2		relres 5965	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dftrrel3 39000	. 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 4568   class class class wbr 5086   ↾ cres 5627  Rel wrel 5630   TrRel wtrrel 38536

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 2709  ax-sep 5232  ax-pr 5371

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-trrel 38996

This theorem is referenced by: (None)