Theorem prtlem16 36165

Description: Lemma for prtex 36176, prter2 36177 and prter3 36178. (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 3444	. . . 4 ⊢ 𝑧 ∈ V
2	1	eldm 5733	. . 3 ⊢ (𝑧 ∈ dom ∼ ↔ ∃𝑤 𝑧 ∼ 𝑤)
3		prtlem13.1	. . . . 5 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
4	3	prtlem13 36164	. . . 4 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
5	4	exbii 1849	. . 3 ⊢ (∃𝑤 𝑧 ∼ 𝑤 ↔ ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
6		elunii 4805	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑣 ∧ 𝑣 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
7	6	ancoms 462	. . . . . . 7 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
8	7	adantrr 716	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑧 ∈ ∪ 𝐴)
9	8	rexlimiva 3240	. . . . 5 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
10	9	exlimiv 1931	. . . 4 ⊢ (∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
11		eluni2 4804	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
12		elequ1 2118	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
13	12	anbi2d 631	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣)))
14		pm4.24 567	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑣 ↔ (𝑧 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
15	13, 14	syl6bbr 292	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧 ∈ 𝑣))
16	15	rexbidv 3256	. . . . . 6 ⊢ (𝑤 = 𝑧 → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣))
17	1, 16	spcev 3555	. . . . 5 ⊢ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 → ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
18	11, 17	sylbi 220	. . . 4 ⊢ (𝑧 ∈ ∪ 𝐴 → ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
19	10, 18	impbii 212	. . 3 ⊢ (∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧 ∈ ∪ 𝐴)
20	2, 5, 19	3bitri 300	. 2 ⊢ (𝑧 ∈ dom ∼ ↔ 𝑧 ∈ ∪ 𝐴)
21	20	eqriv 2795	1 ⊢ dom ∼ = ∪ 𝐴

Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  ∪ cuni 4800   class class class wbr 5030  {copab 5092  dom cdm 5519

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-dm 5529

This theorem is referenced by:  prtlem400  36166  prter1  36175