Theorem bj-elres 38184

Description: Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Use the 𝑥𝐵𝑦 idiom. (Revised by BJ, 16-Oct-2026.) Except for a few early statements (like dfres2 6051 / bj-dfres2 38185), prefer brres 5993.

Assertion

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

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

Proof of Theorem bj-elres

Step	Hyp	Ref	Expression
1		df-res 5670	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
2	1	eleq2i 2856	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ 𝐴 ∈ (𝐵 ∩ (𝐶 × V)))
3		bj-elinxp 38177	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ (𝐶 × V)) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦))
4		rexv 3481	. . 3 ⊢ (∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦))
5	4	rexbii 3113	. 2 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦))
6	2, 3, 5	3bitri 300	1 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ∩ cin 4092  ⟨cop 4840   class class class wbr 5108   × cxp 5656   ↾ cres 5660

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-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-ex 1813  df-sb 2102  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-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-res 5670

This theorem is used by:  bj-dfres2  38185