Theorem ovsng 50300

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 7454	. 2 ⊢ (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = ({⟨⟨𝐴, 𝐵⟩, 𝐶⟩}‘⟨𝐴, 𝐵⟩)
2		opex 5439	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3		fvsng 7216	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → ({⟨⟨𝐴, 𝐵⟩, 𝐶⟩}‘⟨𝐴, 𝐵⟩) = 𝐶)
4	2, 3	mpan 703	. 2 ⊢ (𝐶 ∈ 𝑉 → ({⟨⟨𝐴, 𝐵⟩, 𝐶⟩}‘⟨𝐴, 𝐵⟩) = 𝐶)
5	1, 4	eqtrid 2810	1 ⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454

This theorem is used by:  ovsng2  50301  ovsn  50302