Theorem opelopab3 35875

Description: Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)

Hypotheses

Ref	Expression
opelopab3.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
opelopab3.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
opelopab3.3	⊢ (𝜒 → 𝐴 ∈ 𝐶)

Assertion

Ref	Expression
opelopab3	⊢ (𝐵 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem opelopab3

Step	Hyp	Ref	Expression
1		elopaelxp 5676	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2		opelxp1 5630	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) → 𝐴 ∈ V)
3	1, 2	syl 17	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
4	3	anim1i 615	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
5	4	ancoms 459	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
6		opelopab3.3	. . . . 5 ⊢ (𝜒 → 𝐴 ∈ 𝐶)
7	6	elexd 3452	. . . 4 ⊢ (𝜒 → 𝐴 ∈ V)
8	7	anim1i 615	. . 3 ⊢ ((𝜒 ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
9	8	ancoms 459	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝜒) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
10		opelopab3.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
11		opelopab3.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
12	10, 11	opelopabg 5451	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
13	5, 9, 12	pm5.21nd 799	1 ⊢ (𝐵 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567  {copab 5136   × cxp 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595

This theorem is referenced by: (None)