Theorem prtlem16 39624

Description: Lemma for prtex 39635, prter2 39636 and prter3 39637. (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 3459	. . . 4 ⊢ 𝑧 ∈ V
2	1	eldm 5892	. . 3 ⊢ (𝑧 ∈ dom ∼ ↔ ∃𝑤 𝑧 ∼ 𝑤)
3		prtlem13.1	. . . . 5 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
4	3	prtlem13 39623	. . . 4 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
5	4	exbii 1878	. . 3 ⊢ (∃𝑤 𝑧 ∼ 𝑤 ↔ ∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
6		elunii 4878	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑣 ∧ 𝑣 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
7	6	ancoms 463	. . . . . . 7 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
8	7	adantrr 729	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → 𝑧 ∈ ∪ 𝐴)
9	8	rexlimiva 3158	. . . . 5 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
10	9	exlimiv 1960	. . . 4 ⊢ (∃𝑤∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → 𝑧 ∈ ∪ 𝐴)
11		eluni2 4877	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
12		elequ1 2150	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
13	12	anbi2d 641	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣)))
14		pm4.24 573	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑣 ↔ (𝑧 ∈ 𝑣 ∧ 𝑧 ∈ 𝑣))
15	13, 14	bitr4di 292	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧 ∈ 𝑣))
16	15	rexbidv 3189	. . . . . 6 ⊢ (𝑤 = 𝑧 → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣))
17	1, 16	spcev 3566	. . . . 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 2760	1 ⊢ dom ∼ = ∪ 𝐴

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  ∪ cuni 4873   class class class wbr 5110  {copab 5174  dom cdm 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-dm 5673

This theorem is referenced by:  prtlem400  39625  prter1  39634