Theorem elsnres 5986

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 5985	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2		rexcom4 3264	. 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 4816	. . . . . 6 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
5	4	eqeq2d 2747	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝐶, 𝑦⟩))
6	4	eleq1d 2821	. . . . 5 ⊢ (𝑥 = 𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
7	5, 6	anbi12d 633	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵)))
8	3, 7	rexsn 4626	. . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
9	8	exbii 1850	. 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 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429  {csn 4567  ⟨cop 4573   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-res 5643

This theorem is referenced by:  fvn0ssdmfun  7026  frxp  8076  gsumhashmul  33128