Theorem prtlem16 34473

Description: Lemma for prtex 34484, prter2 34485 and prter3 34486. (Contributed by Rodolfo Medina, 14-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
prtlem16	⊢ dom ∼ = ∪ 𝐴

Distinct variable group:   𝑥,𝑢,𝑦,𝐴

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑢)

Proof of Theorem prtlem16

Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3234	. . . 4 ⊢ 𝑧 ∈ V
2	1	eldm 5353	. . 3 ⊢ (𝑧 ∈ dom ∼ ↔ ∃𝑤 𝑧 ∼ 𝑤)
3		prtlem13.1	. . . . 5 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
4	3	prtlem13 34472	. . . 4 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
5	4	exbii 1814	. . 3 ⊢ (∃𝑤 𝑧 ∼ 𝑤 ↔ ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
6		elunii 4473	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑣 ∧ 𝑣 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
7	6	ancoms 468	. . . . . . 7 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
8	7	adantrr 753	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑧 ∈ ∪ 𝐴)
9	8	rexlimiva 3057	. . . . 5 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
10	9	exlimiv 1898	. . . 4 ⊢ (∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
11		eluni2 4472	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
12		eleq1 2718	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
13	12	anbi2d 740	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣)))
14		pm4.24 676	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑣 ↔ (𝑧 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
15	13, 14	syl6bbr 278	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧 ∈ 𝑣))
16	15	rexbidv 3081	. . . . . 6 ⊢ (𝑤 = 𝑧 → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣))
17	1, 16	spcev 3331	. . . . 5 ⊢ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 → ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
18	11, 17	sylbi 207	. . . 4 ⊢ (𝑧 ∈ ∪ 𝐴 → ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
19	10, 18	impbii 199	. . 3 ⊢ (∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧 ∈ ∪ 𝐴)
20	2, 5, 19	3bitri 286	. 2 ⊢ (𝑧 ∈ dom ∼ ↔ 𝑧 ∈ ∪ 𝐴)
21	20	eqriv 2648	1 ⊢ dom ∼ = ∪ 𝐴

Syntax hints:   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃wrex 2942  ∪ cuni 4468   class class class wbr 4685  {copab 4745  dom cdm 5143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-dm 5153

This theorem is referenced by:  prtlem400  34474  prter1  34483