Theorem prtlem16 39125

Description: Lemma for prtex 39136, prter2 39137 and prter3 39138. (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 5849	. . 3 ⊢ (𝑧 ∈ dom ∼ ↔ ∃𝑤 𝑧 ∼ 𝑤)
3		prtlem13.1	. . . . 5 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
4	3	prtlem13 39124	. . . 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 4868	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑣 ∧ 𝑣 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
7	6	ancoms 458	. . . . . . 7 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
8	7	adantrr 717	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑧 ∈ ∪ 𝐴)
9	8	rexlimiva 3129	. . . . 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 4867	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
12		elequ1 2120	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
13	12	anbi2d 630	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣)))
14		pm4.24 563	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑣 ↔ (𝑧 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
15	13, 14	bitr4di 289	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧 ∈ 𝑣))
16	15	rexbidv 3160	. . . . . 6 ⊢ (𝑤 = 𝑧 → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣))
17	1, 16	spcev 3560	. . . . 5 ⊢ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 → ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
18	11, 17	sylbi 217	. . . 4 ⊢ (𝑧 ∈ ∪ 𝐴 → ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
19	10, 18	impbii 209	. . 3 ⊢ (∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧 ∈ ∪ 𝐴)
20	2, 5, 19	3bitri 297	. 2 ⊢ (𝑧 ∈ dom ∼ ↔ 𝑧 ∈ ∪ 𝐴)
21	20	eqriv 2733	1 ⊢ dom ∼ = ∪ 𝐴

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3060  ∪ cuni 4863   class class class wbr 5098  {copab 5160  dom cdm 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-dm 5634

This theorem is referenced by:  prtlem400  39126  prter1  39135