Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem16 Structured version   Visualization version   GIF version

Theorem prtlem16 35885
Description: Lemma for prtex 35896, prter2 35897 and prter3 35898. (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 3495 . . . 4 𝑧 ∈ V
21eldm 5762 . . 3 (𝑧 ∈ dom ↔ ∃𝑤 𝑧 𝑤)
3 prtlem13.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43prtlem13 35884 . . . 4 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
54exbii 1839 . . 3 (∃𝑤 𝑧 𝑤 ↔ ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
6 elunii 4835 . . . . . . . 8 ((𝑧𝑣𝑣𝐴) → 𝑧 𝐴)
76ancoms 459 . . . . . . 7 ((𝑣𝐴𝑧𝑣) → 𝑧 𝐴)
87adantrr 713 . . . . . 6 ((𝑣𝐴 ∧ (𝑧𝑣𝑤𝑣)) → 𝑧 𝐴)
98rexlimiva 3278 . . . . 5 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
109exlimiv 1922 . . . 4 (∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
11 eluni2 4834 . . . . 5 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
12 elequ1 2112 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤𝑣𝑧𝑣))
1312anbi2d 628 . . . . . . . 8 (𝑤 = 𝑧 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑣𝑧𝑣)))
14 pm4.24 564 . . . . . . . 8 (𝑧𝑣 ↔ (𝑧𝑣𝑧𝑣))
1513, 14syl6bbr 290 . . . . . . 7 (𝑤 = 𝑧 → ((𝑧𝑣𝑤𝑣) ↔ 𝑧𝑣))
1615rexbidv 3294 . . . . . 6 (𝑤 = 𝑧 → (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ ∃𝑣𝐴 𝑧𝑣))
171, 16spcev 3604 . . . . 5 (∃𝑣𝐴 𝑧𝑣 → ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
1811, 17sylbi 218 . . . 4 (𝑧 𝐴 → ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
1910, 18impbii 210 . . 3 (∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ 𝑧 𝐴)
202, 5, 193bitri 298 . 2 (𝑧 ∈ dom 𝑧 𝐴)
2120eqriv 2815 1 dom = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136   cuni 4830   class class class wbr 5057  {copab 5119  dom cdm 5548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-dm 5558
This theorem is referenced by:  prtlem400  35886  prter1  35895
  Copyright terms: Public domain W3C validator