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Theorem trressn 37315
Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37312) is transitive, see also trrelressn 37453. (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 658 . . . . 5 (((𝑥 = 𝐴𝐴𝑅𝑦) ∧ (𝑦 = 𝐴𝐴𝑅𝑧)) → (𝑥 = 𝐴𝐴𝑅𝑧))
2 eqbrb 37099 . . . . . 6 ((𝑥 = 𝐴𝑥𝑅𝑦) ↔ (𝑥 = 𝐴𝐴𝑅𝑦))
3 eqbrb 37099 . . . . . 6 ((𝑦 = 𝐴𝑦𝑅𝑧) ↔ (𝑦 = 𝐴𝐴𝑅𝑧))
42, 3anbi12i 628 . . . . 5 (((𝑥 = 𝐴𝑥𝑅𝑦) ∧ (𝑦 = 𝐴𝑦𝑅𝑧)) ↔ ((𝑥 = 𝐴𝐴𝑅𝑦) ∧ (𝑦 = 𝐴𝐴𝑅𝑧)))
5 eqbrb 37099 . . . . 5 ((𝑥 = 𝐴𝑥𝑅𝑧) ↔ (𝑥 = 𝐴𝐴𝑅𝑧))
61, 4, 53imtr4i 292 . . . 4 (((𝑥 = 𝐴𝑥𝑅𝑦) ∧ (𝑦 = 𝐴𝑦𝑅𝑧)) → (𝑥 = 𝐴𝑥𝑅𝑧))
7 brressn 37311 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦)))
87el2v 3483 . . . . 5 (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴𝑥𝑅𝑦))
9 brressn 37311 . . . . . 6 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴𝑦𝑅𝑧)))
109el2v 3483 . . . . 5 (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴𝑦𝑅𝑧))
118, 10anbi12i 628 . . . 4 ((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) ↔ ((𝑥 = 𝐴𝑥𝑅𝑦) ∧ (𝑦 = 𝐴𝑦𝑅𝑧)))
12 brressn 37311 . . . . 5 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴𝑥𝑅𝑧)))
1312el2v 3483 . . . 4 (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴𝑥𝑅𝑧))
146, 11, 133imtr4i 292 . . 3 ((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
1514gen2 1799 . 2 𝑦𝑧((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
1615ax-gen 1798 1 𝑥𝑦𝑧((𝑥(𝑅 ↾ {𝐴})𝑦𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  Vcvv 3475  {csn 4629   class class class wbr 5149  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-res 5689
This theorem is referenced by:  trrelressn  37453
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