Theorem opelres 6006

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 5701	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
2	1	elin2 4213	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)))
3		opelxp 5725	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V))
4		elex 3499	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
5	4	biantrud 531	. . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝐵 ∈ 𝐴 ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V)))
6	3, 5	bitr4id 290	. . 3 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ 𝐵 ∈ 𝐴))
7	6	anbi1cd 635	. 2 ⊢ (𝐶 ∈ 𝑉 → ((⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
8	2, 7	bitrid 283	1 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  Vcvv 3478  ⟨cop 4637   × cxp 5687   ↾ cres 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-res 5701

This theorem is referenced by:  brres  6007  opelresi  6008  opelidres  6012  h2hlm  31009  setsnidel  47302