Theorem elsnres 5972

Description: Membership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)

Hypothesis

Ref	Expression
elsnres.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elsnres	⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶

Proof of Theorem elsnres

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elres 5971	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2		rexcom4 3256	. 2 ⊢ (∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		elsnres.1	. . . 4 ⊢ 𝐶 ∈ V
4		opeq1 4824	. . . . . 6 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
5	4	eqeq2d 2740	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝐶, 𝑦⟩))
6	4	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
7	5, 6	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵)))
8	3, 7	rexsn 4634	. . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
9	8	exbii 1848	. 2 ⊢ (∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
10	1, 2, 9	3bitri 297	1 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  Vcvv 3436  {csn 4577  ⟨cop 4583   ↾ cres 5621

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-11 2158  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-res 5631

This theorem is referenced by:  fvn0ssdmfun  7008  frxp  8059  gsumhashmul  33015