Theorem prtlem13 38251

Description: Lemma for prter1 38262, prter2 38264, prter3 38265 and prtex 38263. (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 3472	. 2 ⊢ 𝑧 ∈ V
2		vex 3472	. 2 ⊢ 𝑤 ∈ V
3		elequ2 2113	. . . . 5 ⊢ (𝑢 = 𝑣 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑣))
4		elequ2 2113	. . . . 5 ⊢ (𝑢 = 𝑣 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑣))
5	3, 4	anbi12d 630	. . . 4 ⊢ (𝑢 = 𝑣 → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣)))
6	5	cbvrexvw 3229	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑣 ∈ 𝐴 (𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣))
7		elequ1 2105	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
8		elequ1 2105	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑤 ∈ 𝑣))
9	7, 8	bi2anan9 636	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝑣 ∧ 𝑦 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
10	9	rexbidv 3172	. . 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 5530	1 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wrex 3064   class class class wbr 5141  {copab 5203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204

This theorem is referenced by:  prtlem16  38252  prtlem18  38260  prter1  38262  prter3  38265