Theorem opelres 5831

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 5537	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
2	1	elin2 4087	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)))
3		opelxp 5561	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V))
4		elex 3416	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
5	4	biantrud 535	. . . 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 637	. 2 ⊢ (𝐶 ∈ 𝑉 → ((⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
8	2, 7	syl5bb 286	1 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2114  Vcvv 3398  ⟨cop 4522   × cxp 5523   ↾ cres 5527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-opab 5093  df-xp 5531  df-res 5537

This theorem is referenced by:  brres  5832  opelresi  5833  opelidres  5837  h2hlm  28915  setsnidel  44363