Theorem ovsng 49019

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

Assertion

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

Proof of Theorem ovsng

Step	Hyp	Ref	Expression
1		df-ov 7358	. 2 ⊢ (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = ({⟨⟨𝐴, 𝐵⟩, 𝐶⟩}‘⟨𝐴, 𝐵⟩)
2		opex 5409	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3		fvsng 7123	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → ({⟨⟨𝐴, 𝐵⟩, 𝐶⟩}‘⟨𝐴, 𝐵⟩) = 𝐶)
4	2, 3	mpan 690	. 2 ⊢ (𝐶 ∈ 𝑉 → ({⟨⟨𝐴, 𝐵⟩, 𝐶⟩}‘⟨𝐴, 𝐵⟩) = 𝐶)
5	1, 4	eqtrid 2780	1 ⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  {csn 4577  ⟨cop 4583  ‘cfv 6489  (class class class)co 7355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  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 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358

This theorem is referenced by:  ovsng2  49020  ovsn  49021