Theorem opelres 5981

Description: Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) (Revised by BJ, 18-Feb-2022.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.)

Assertion

Ref	Expression
opelres	⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))

Proof of Theorem opelres

Step	Hyp	Ref	Expression
1		df-res 5667	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
2	1	elin2 4103	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)))
3		opelxp 5691	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V))
4		elex 3475	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
5	4	biantrud 541	. . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝐵 ∈ 𝐴 ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V)))
6	3, 5	bitr4id 293	. . 3 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ 𝐵 ∈ 𝐴))
7	6	anbi1cd 647	. 2 ⊢ (𝐶 ∈ 𝑉 → ((⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
8	2, 7	bitrid 286	1 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  Vcvv 3454  ⟨cop 4838   × cxp 5653   ↾ cres 5657

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  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-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-res 5667

This theorem is used by:  brres  5982  opelresi  5983  opelidres  5987  h2hlm  31597  setsnidel  48601