Theorem rabxp 5603

Description: Membership in a class builder restricted to a Cartesian product. (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 5581	. . . . 5 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
2	1	anbi1i 625	. . . 4 ⊢ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
3		19.41vv 1950	. . . 4 ⊢ (∃𝑦∃𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
4		anass 471	. . . . . 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 630	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜓)))
7		df-3an 1085	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜓))
8	6, 7	syl6bbr 291	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
9	8	pm5.32i 577	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
10	4, 9	bitri 277	. . . . 5 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
11	10	2exbii 1848	. . . 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 301	. . 3 ⊢ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)))
13	12	abbii 2889	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {𝑥 ∣ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓))}
14		df-rab 3150	. 2 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑)}
15		df-opab 5132	. 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)} = {𝑥 ∣ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓))}
16	13, 14, 15	3eqtr4i 2857	1 ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)}

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2802  {crab 3145  ⟨cop 4576  {copab 5131   × cxp 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-opab 5132  df-xp 5564

This theorem is referenced by:  cicer  17079  poimirlem26  34922  dib1dim  38305  diclspsn  38334  fgraphxp  39817