Theorem trressn 37363

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37360) is transitive, see also trrelressn 37501. (Contributed by Peter Mazsa, 16-Jun-2024.)

Assertion

Ref	Expression
trressn	⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)

Proof of Theorem trressn

Step	Hyp	Ref	Expression
1		an3 658	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝐴𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧)) → (𝑥 = 𝐴 ∧ 𝐴𝑅𝑧))
2		eqbrb 37147	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑦))
3		eqbrb 37147	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑦𝑅𝑧) ↔ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧))
4	2, 3	anbi12i 628	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) ↔ ((𝑥 = 𝐴 ∧ 𝐴𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧)))
5		eqbrb 37147	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑧) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑧))
6	1, 4, 5	3imtr4i 292	. . . 4 ⊢ (((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧))
7		brressn 37359	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)))
8	7	el2v 3483	. . . . 5 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))
9		brressn 37359	. . . . . 6 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)))
10	9	el2v 3483	. . . . 5 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧))
11	8, 10	anbi12i 628	. . . 4 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) ↔ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)))
12		brressn 37359	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧)))
13	12	el2v 3483	. . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧))
14	6, 11, 13	3imtr4i 292	. . 3 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
15	14	gen2 1799	. 2 ⊢ ∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
16	15	ax-gen 1798	1 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)

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  37501