Theorem opelopab3 38229

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 5742	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2		opelxp1 5694	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) → 𝐴 ∈ V)
3	1, 2	syl 18	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
4	3	anim1i 626	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
5	4	ancoms 463	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
6		opelopab3.3	. . . . 5 ⊢ (𝜒 → 𝐴 ∈ 𝐶)
7	6	elexd 3480	. . . 4 ⊢ (𝜒 → 𝐴 ∈ V)
8	7	anim1i 626	. . 3 ⊢ ((𝜒 ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
9	8	ancoms 463	. 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 5514	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
13	5, 9, 12	pm5.21nd 813	1 ⊢ (𝐵 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ⟨cop 4591  {copab 5167   × cxp 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5168  df-xp 5658

This theorem is referenced by: (None)