Theorem trrelressn 38564

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38423) 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 38426	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
2		relres 5956	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dftrrel3 38559	. 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 711	1 ⊢ TrRel (𝑅 ↾ {𝐴})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  {csn 4577   class class class wbr 5092   ↾ cres 5621  Rel wrel 5624   TrRel wtrrel 38174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-trrel 38555

This theorem is referenced by: (None)