Theorem opelresg 5020

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 3863	. . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2	1	eleq1d 2300	. 2 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷)))
3	1	eleq1d 2300	. . 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 2805	. . 3 ⊢ 𝑦 ∈ V
6	5	opelres 5018	. 2 ⊢ (⟨𝐴, 𝑦⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))
7	2, 4, 6	vtoclbg 2865	1 ⊢ (𝐵 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ⟨cop 3672   ↾ cres 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-res 4737

This theorem is referenced by:  brresg  5021  opelresi  5024  issref  5119