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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⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
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.
StepHypRef Expression
1 vex 3234 . . . 4 𝑧 ∈ V
21eldm 5353 . . 3 (𝑧 ∈ dom ↔ ∃𝑤 𝑧 𝑤)
3 prtlem13.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43prtlem13 34472 . . . 4 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
54exbii 1814 . . 3 (∃𝑤 𝑧 𝑤 ↔ ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
6 elunii 4473 . . . . . . . 8 ((𝑧𝑣𝑣𝐴) → 𝑧 𝐴)
76ancoms 468 . . . . . . 7 ((𝑣𝐴𝑧𝑣) → 𝑧 𝐴)
87adantrr 753 . . . . . 6 ((𝑣𝐴 ∧ (𝑧𝑣𝑤𝑣)) → 𝑧 𝐴)
98rexlimiva 3057 . . . . 5 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
109exlimiv 1898 . . . 4 (∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
11 eluni2 4472 . . . . 5 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
12 eleq1 2718 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤𝑣𝑧𝑣))
1312anbi2d 740 . . . . . . . 8 (𝑤 = 𝑧 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑣𝑧𝑣)))
14 pm4.24 676 . . . . . . . 8 (𝑧𝑣 ↔ (𝑧𝑣𝑧𝑣))
1513, 14syl6bbr 278 . . . . . . 7 (𝑤 = 𝑧 → ((𝑧𝑣𝑤𝑣) ↔ 𝑧𝑣))
1615rexbidv 3081 . . . . . 6 (𝑤 = 𝑧 → (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ ∃𝑣𝐴 𝑧𝑣))
171, 16spcev 3331 . . . . 5 (∃𝑣𝐴 𝑧𝑣 → ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
1811, 17sylbi 207 . . . 4 (𝑧 𝐴 → ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
1910, 18impbii 199 . . 3 (∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ 𝑧 𝐴)
202, 5, 193bitri 286 . 2 (𝑧 ∈ dom 𝑧 𝐴)
2120eqriv 2648 1 dom = 𝐴
Colors of variables: wff setvar class
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
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