Theorem bj-elres 38057

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.)

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 5663	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
2	1	eleq2i 2852	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ 𝐴 ∈ (𝐵 ∩ (𝐶 × V)))
3		bj-elinxp 38056	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ (𝐶 × V)) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦))
4		rexv 3477	. . 3 ⊢ (∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦))
5	4	rexbii 3109	. 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 1569  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ∩ cin 4088  ⟨cop 4834   class class class wbr 5102   × cxp 5649   ↾ cres 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-un 4150  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-res 5663

This theorem is used by:  bj-dfres2  38058