Theorem trressn 39034

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 39031) is transitive, see also trrelressn 39166. (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 669	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝐴𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧)) → (𝑥 = 𝐴 ∧ 𝐴𝑅𝑧))
2		eqbrb 38738	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑦))
3		eqbrb 38738	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑦𝑅𝑧) ↔ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧))
4	2, 3	anbi12i 637	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) ↔ ((𝑥 = 𝐴 ∧ 𝐴𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧)))
5		eqbrb 38738	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑧) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑧))
6	1, 4, 5	3imtr4i 294	. . . 4 ⊢ (((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧))
7		brressn 39030	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)))
8	7	el2v 3461	. . . . 5 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))
9		brressn 39030	. . . . . 6 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)))
10	9	el2v 3461	. . . . 5 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧))
11	8, 10	anbi12i 637	. . . 4 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) ↔ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)))
12		brressn 39030	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧)))
13	12	el2v 3461	. . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧))
14	6, 11, 13	3imtr4i 294	. . 3 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
15	14	gen2 1816	. 2 ⊢ ∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
16	15	ax-gen 1815	1 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560  Vcvv 3454  {csn 4582   class class class wbr 5100   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-res 5659

This theorem is referenced by:  trrelressn  39166