Theorem opelresg 5324

Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)

Assertion

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

Proof of Theorem opelresg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4341	. . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2	1	eleq1d 2672	. 2 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷)))
3	1	eleq1d 2672	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
4	3	anbi1d 737	. 2 ⊢ (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)))
5		vex 3176	. . 3 ⊢ 𝑦 ∈ V
6	5	opelres 5322	. 2 ⊢ (⟨𝐴, 𝑦⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))
7	2, 4, 6	vtoclbg 3240	1 ⊢ (𝐵 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   ↾ cres 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-res 5050

This theorem is referenced by:  brresg  5325  opelresi  5328  issref  5428