Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trressn Structured version   Visualization version   GIF version

Theorem trressn 38427
Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38424) is transitive, see also trrelressn 38565. (Contributed by Peter Mazsa, 16-Jun-2024.)
Assertion
Ref Expression
trressn 𝑥𝑦𝑧((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)

Proof of Theorem trressn
StepHypRef Expression
1 an3 659 . . . . 5 (((𝑥 = 𝐴𝐴𝑅𝑦) ∧ (𝑦 = 𝐴𝐴𝑅𝑧)) → (𝑥 = 𝐴𝐴𝑅𝑧))
2 eqbrb 38214 . . . . . 6 ((𝑥 = 𝐴𝑥𝑅𝑦) ↔ (𝑥 = 𝐴𝐴𝑅𝑦))
3 eqbrb 38214 . . . . . 6 ((𝑦 = 𝐴𝑦𝑅𝑧) ↔ (𝑦 = 𝐴𝐴𝑅𝑧))
42, 3anbi12i 628 . . . . 5 (((𝑥 = 𝐴𝑥𝑅𝑦) ∧ (𝑦 = 𝐴𝑦𝑅𝑧)) ↔ ((𝑥 = 𝐴𝐴𝑅𝑦) ∧ (𝑦 = 𝐴𝐴𝑅𝑧)))
5 eqbrb 38214 . . . . 5 ((𝑥 = 𝐴𝑥𝑅𝑧) ↔ (𝑥 = 𝐴𝐴𝑅𝑧))
61, 4, 53imtr4i 292 . . . 4 (((𝑥 = 𝐴𝑥𝑅𝑦) ∧ (𝑦 = 𝐴𝑦𝑅𝑧)) → (𝑥 = 𝐴𝑥𝑅𝑧))
7 brressn 38423 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦)))
87el2v 3485 . . . . 5 (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦))
9 brressn 38423 . . . . . 6 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴𝑦𝑅𝑧)))
109el2v 3485 . . . . 5 (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴𝑦𝑅𝑧))
118, 10anbi12i 628 . . . 4 ((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) ↔ ((𝑥 = 𝐴𝑥𝑅𝑦) ∧ (𝑦 = 𝐴𝑦𝑅𝑧)))
12 brressn 38423 . . . . 5 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴𝑥𝑅𝑧)))
1312el2v 3485 . . . 4 (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴𝑥𝑅𝑧))
146, 11, 133imtr4i 292 . . 3 ((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
1514gen2 1793 . 2 𝑦𝑧((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
1615ax-gen 1792 1 𝑥𝑦𝑧((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  Vcvv 3478  {csn 4631   class class class wbr 5148  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701
This theorem is referenced by:  trrelressn  38565
  Copyright terms: Public domain W3C validator