Theorem elres 5976

Description: Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.) (Proof shortened by Peter Mazsa, 9-Sep-2022.)

Assertion

Ref	Expression
elres	⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

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

Proof of Theorem elres

Step	Hyp	Ref	Expression
1		df-res 5633	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
2	1	eleq2i 2825	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ 𝐴 ∈ (𝐵 ∩ (𝐶 × V)))
3		elinxp 5975	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ (𝐶 × V)) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
4		rexv 3465	. . 3 ⊢ (∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	rexbii 3080	. 2 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6	2, 3, 5	3bitri 297	1 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3057  Vcvv 3437   ∩ cin 3897  ⟨cop 4583   × cxp 5619   ↾ cres 5623

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-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-sb 2068  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-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-res 5633

This theorem is referenced by:  elsnres  5977  eldm3  35877