Theorem ovsng2 48743

Description: The operation value of a singleton of an ordered triple is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)

Assertion

Ref	Expression
ovsng2	⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = 𝐶)

Proof of Theorem ovsng2

Step	Hyp	Ref	Expression
1		df-ot 4615	. . . 4 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	sneqi 4617	. . 3 ⊢ {⟨𝐴, 𝐵, 𝐶⟩} = {⟨⟨𝐴, 𝐵⟩, 𝐶⟩}
3	2	oveqi 7426	. 2 ⊢ (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵)
4		ovsng 48742	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶)
5	3, 4	eqtrid 2781	1 ⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = 𝐶)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  {csn 4606  ⟨cop 4612  ⟨cotp 4614  (class class class)co 7413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-ot 4615  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416

This theorem is referenced by:  ovsn2  48745