Theorem ovsng2 48863

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 4588	. . . 4 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	sneqi 4590	. . 3 ⊢ {⟨𝐴, 𝐵, 𝐶⟩} = {⟨⟨𝐴, 𝐵⟩, 𝐶⟩}
3	2	oveqi 7366	. 2 ⊢ (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵)
4		ovsng 48862	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶)
5	3, 4	eqtrid 2776	1 ⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = 𝐶)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4579  ⟨cop 4585  ⟨cotp 4587  (class class class)co 7353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356

This theorem is referenced by:  ovsn2  48865