Theorem trressn 39467

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 39464) is transitive, see also trrelressn 39599. (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 672	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝐴𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧)) → (𝑥 = 𝐴 ∧ 𝐴𝑅𝑧))
2		eqbrb 39171	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑦))
3		eqbrb 39171	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑦𝑅𝑧) ↔ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧))
4	2, 3	anbi12i 640	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) ↔ ((𝑥 = 𝐴 ∧ 𝐴𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧)))
5		eqbrb 39171	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑧) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑧))
6	1, 4, 5	3imtr4i 295	. . . 4 ⊢ (((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧))
7		brressn 39463	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)))
8	7	el2v 3458	. . . . 5 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))
9		brressn 39463	. . . . . 6 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)))
10	9	el2v 3458	. . . . 5 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧))
11	8, 10	anbi12i 640	. . . 4 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) ↔ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)))
12		brressn 39463	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧)))
13	12	el2v 3458	. . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧))
14	6, 11, 13	3imtr4i 295	. . 3 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
15	14	gen2 1829	. 2 ⊢ ∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
16	15	ax-gen 1828	1 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  Vcvv 3451  {csn 4584   class class class wbr 5103   ↾ cres 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-res 5663

This theorem is used by:  trrelressn  39599