Theorem prtlem13 39241

Description: Lemma for prter1 39252, prter2 39254, prter3 39255 and prtex 39253. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem13.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}

Assertion

Ref	Expression
prtlem13	⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))

Distinct variable groups:   𝑣,𝑢,𝑥,𝑦,𝐴   𝑤,𝑣,𝑥,𝑦   𝑧,𝑣,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑧,𝑤)   ∼ (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem prtlem13

Step	Hyp	Ref	Expression
1		vex 3446	. 2 ⊢ 𝑧 ∈ V
2		vex 3446	. 2 ⊢ 𝑤 ∈ V
3		elequ2 2129	. . . . 5 ⊢ (𝑢 = 𝑣 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑣))
4		elequ2 2129	. . . . 5 ⊢ (𝑢 = 𝑣 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑣))
5	3, 4	anbi12d 633	. . . 4 ⊢ (𝑢 = 𝑣 → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣)))
6	5	cbvrexvw 3217	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑣 ∈ 𝐴 (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣))
7		elequ1 2121	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
8		elequ1 2121	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑤 ∈ 𝑣))
9	7, 8	bi2anan9 639	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
10	9	rexbidv 3162	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑣 ∈ 𝐴 (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
11	6, 10	bitrid 283	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
12		prtlem13.1	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
13	1, 2, 11, 12	braba 5493	1 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wrex 3062   class class class wbr 5100  {copab 5162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163

This theorem is referenced by:  prtlem16  39242  prtlem18  39250  prter1  39252  prter3  39255