Theorem opelopab3 38181

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 5735	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2		opelxp1 5687	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) → 𝐴 ∈ V)
3	1, 2	syl 17	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
4	3	anim1i 624	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
5	4	ancoms 462	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
6		opelopab3.3	. . . . 5 ⊢ (𝜒 → 𝐴 ∈ 𝐶)
7	6	elexd 3476	. . . 4 ⊢ (𝜒 → 𝐴 ∈ V)
8	7	anim1i 624	. . 3 ⊢ ((𝜒 ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷))
9	8	ancoms 462	. 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 5507	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
13	5, 9, 12	pm5.21nd 811	1 ⊢ (𝐵 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ⟨cop 4587  {copab 5161   × cxp 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-opab 5162  df-xp 5651

This theorem is referenced by: (None)