Theorem opelresg 4985

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 3834	. . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2	1	eleq1d 2276	. 2 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷)))
3	1	eleq1d 2276	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
4	3	anbi1d 465	. 2 ⊢ (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)))
5		vex 2779	. . 3 ⊢ 𝑦 ∈ V
6	5	opelres 4983	. 2 ⊢ (⟨𝐴, 𝑦⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))
7	2, 4, 6	vtoclbg 2839	1 ⊢ (𝐵 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2178  ⟨cop 3646   ↾ cres 4695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-xp 4699  df-res 4705

This theorem is referenced by:  brresg  4986  opelresi  4989  issref  5084