Theorem elres 6020

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 5674	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
2	1	eleq2i 2861	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ 𝐴 ∈ (𝐵 ∩ (𝐶 × V)))
3		elinxp 6019	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ (𝐶 × V)) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
4		rexv 3490	. . 3 ⊢ (∃𝑦 ∈ V (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	rexbii 3118	. 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 300	1 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463   ∩ cin 3912  ⟨cop 4600   × cxp 5660   ↾ cres 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-res 5674

This theorem is referenced by:  elsnres  6021  eldm3  36151