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Theorem prtlem16 33172
 Description: Lemma for prtex 33183, prter2 33184 and prter3 33185. (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 3176 . . . 4 𝑧 ∈ V
21eldm 5243 . . 3 (𝑧 ∈ dom ↔ ∃𝑤 𝑧 𝑤)
3 prtlem13.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43prtlem13 33171 . . . 4 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
54exbii 1764 . . 3 (∃𝑤 𝑧 𝑤 ↔ ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
6 elunii 4377 . . . . . . . 8 ((𝑧𝑣𝑣𝐴) → 𝑧 𝐴)
76ancoms 468 . . . . . . 7 ((𝑣𝐴𝑧𝑣) → 𝑧 𝐴)
87adantrr 749 . . . . . 6 ((𝑣𝐴 ∧ (𝑧𝑣𝑤𝑣)) → 𝑧 𝐴)
98rexlimiva 3010 . . . . 5 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
109exlimiv 1845 . . . 4 (∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
11 eluni2 4376 . . . . 5 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
12 eleq1 2676 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤𝑣𝑧𝑣))
1312anbi2d 736 . . . . . . . 8 (𝑤 = 𝑧 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑣𝑧𝑣)))
14 pm4.24 673 . . . . . . . 8 (𝑧𝑣 ↔ (𝑧𝑣𝑧𝑣))
1513, 14syl6bbr 277 . . . . . . 7 (𝑤 = 𝑧 → ((𝑧𝑣𝑤𝑣) ↔ 𝑧𝑣))
1615rexbidv 3034 . . . . . 6 (𝑤 = 𝑧 → (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ ∃𝑣𝐴 𝑧𝑣))
171, 16spcev 3273 . . . . 5 (∃𝑣𝐴 𝑧𝑣 → ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
1811, 17sylbi 206 . . . 4 (𝑧 𝐴 → ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
1910, 18impbii 198 . . 3 (∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ 𝑧 𝐴)
202, 5, 193bitri 285 . 2 (𝑧 ∈ dom 𝑧 𝐴)
2120eqriv 2607 1 dom = 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  ∪ cuni 4372   class class class wbr 4583  {copab 4642  dom cdm 5038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-dm 5048 This theorem is referenced by:  prtlem400  33173  prter1  33182
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