Theorem trrelressn 39034

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38899) 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 38902	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
2		relres 5957	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dftrrel3 39029	. 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 717	1 ⊢ TrRel (𝑅 ↾ {𝐴})

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545  {csn 4555   class class class wbr 5072   ↾ cres 5620  Rel wrel 5623   TrRel wtrrel 38565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-trrel 39025

This theorem is referenced by: (None)