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Theorem prtlem16 35873
Description: Lemma for prtex 35884, prter2 35885 and prter3 35886. (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 3502 . . . 4 𝑧 ∈ V
21eldm 5767 . . 3 (𝑧 ∈ dom ↔ ∃𝑤 𝑧 𝑤)
3 prtlem13.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43prtlem13 35872 . . . 4 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
54exbii 1841 . . 3 (∃𝑤 𝑧 𝑤 ↔ ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
6 elunii 4841 . . . . . . . 8 ((𝑧𝑣𝑣𝐴) → 𝑧 𝐴)
76ancoms 459 . . . . . . 7 ((𝑣𝐴𝑧𝑣) → 𝑧 𝐴)
87adantrr 713 . . . . . 6 ((𝑣𝐴 ∧ (𝑧𝑣𝑤𝑣)) → 𝑧 𝐴)
98rexlimiva 3285 . . . . 5 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
109exlimiv 1924 . . . 4 (∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
11 eluni2 4840 . . . . 5 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
12 elequ1 2113 . . . . . . . . 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 3301 . . . . . 6 (𝑤 = 𝑧 → (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ ∃𝑣𝐴 𝑧𝑣))
171, 16spcev 3610 . . . . 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 2821 1 dom = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1530  wex 1773  wcel 2106  wrex 3143   cuni 4836   class class class wbr 5062  {copab 5124  dom cdm 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-dm 5563
This theorem is referenced by:  prtlem400  35874  prter1  35883
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