Theorem rabxp 5634

Description: Class abstraction restricted to a Cartesian product as an ordered-pair class abstraction. (Contributed by NM, 20-Feb-2014.)

Hypothesis

Ref	Expression
rabxp.1	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rabxp	⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)}

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝜑,𝑦,𝑧   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem rabxp

Step	Hyp	Ref	Expression
1		elxp 5611	. . . . 5 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
2	1	anbi1i 623	. . . 4 ⊢ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
3		19.41vv 1957	. . . 4 ⊢ (∃𝑦∃𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
4		anass 468	. . . . . 6 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑)))
5		rabxp.1	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
6	5	anbi2d 628	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜓)))
7		df-3an 1087	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜓))
8	6, 7	bitr4di 288	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
9	8	pm5.32i 574	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
10	4, 9	bitri 274	. . . . 5 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
11	10	2exbii 1854	. . . 4 ⊢ (∃𝑦∃𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
12	2, 3, 11	3bitr2i 298	. . 3 ⊢ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
13	12	abbii 2809	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {𝑥 ∣ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓))}
14		df-rab 3074	. 2 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑)}
15		df-opab 5141	. 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)} = {𝑥 ∣ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓))}
16	13, 14, 15	3eqtr4i 2777	1 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541  ∃wex 1785   ∈ wcel 2109  {cab 2716  {crab 3069  ⟨cop 4572  {copab 5140   × cxp 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-opab 5141  df-xp 5594

This theorem is referenced by:  cicer  17499  poimirlem26  35782  dib1dim  39158  diclspsn  39187  fgraphxp  41016