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Theorem prtlem16 33673
Description: Lemma for prtex 33684, prter2 33685 and prter3 33686. (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 3193 . . . 4 𝑧 ∈ V
21eldm 5291 . . 3 (𝑧 ∈ dom ↔ ∃𝑤 𝑧 𝑤)
3 prtlem13.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43prtlem13 33672 . . . 4 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
54exbii 1771 . . 3 (∃𝑤 𝑧 𝑤 ↔ ∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣))
6 elunii 4414 . . . . . . . 8 ((𝑧𝑣𝑣𝐴) → 𝑧 𝐴)
76ancoms 469 . . . . . . 7 ((𝑣𝐴𝑧𝑣) → 𝑧 𝐴)
87adantrr 752 . . . . . 6 ((𝑣𝐴 ∧ (𝑧𝑣𝑤𝑣)) → 𝑧 𝐴)
98rexlimiva 3023 . . . . 5 (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
109exlimiv 1855 . . . 4 (∃𝑤𝑣𝐴 (𝑧𝑣𝑤𝑣) → 𝑧 𝐴)
11 eluni2 4413 . . . . 5 (𝑧 𝐴 ↔ ∃𝑣𝐴 𝑧𝑣)
12 eleq1 2686 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤𝑣𝑧𝑣))
1312anbi2d 739 . . . . . . . 8 (𝑤 = 𝑧 → ((𝑧𝑣𝑤𝑣) ↔ (𝑧𝑣𝑧𝑣)))
14 pm4.24 674 . . . . . . . 8 (𝑧𝑣 ↔ (𝑧𝑣𝑧𝑣))
1513, 14syl6bbr 278 . . . . . . 7 (𝑤 = 𝑧 → ((𝑧𝑣𝑤𝑣) ↔ 𝑧𝑣))
1615rexbidv 3047 . . . . . 6 (𝑤 = 𝑧 → (∃𝑣𝐴 (𝑧𝑣𝑤𝑣) ↔ ∃𝑣𝐴 𝑧𝑣))
171, 16spcev 3290 . . . . 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 2618 1 dom = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2909   cuni 4409   class class class wbr 4623  {copab 4682  dom cdm 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-dm 5094
This theorem is referenced by:  prtlem400  33674  prter1  33683
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