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Theorem prtlem19 38052
Description: Lemma for prter2 38055. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Assertion
Ref Expression
prtlem19 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
Distinct variable groups:   𝑣,𝑢,𝑥,𝑦,𝑧,𝐴   𝑣, ,𝑧
Allowed substitution hints:   (𝑥,𝑦,𝑢)

Proof of Theorem prtlem19
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 prtlem18.1 . . . . . 6 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
21prtlem18 38051 . . . . 5 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
32imp 406 . . . 4 ((Prt 𝐴 ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑧 𝑤))
4 vex 3477 . . . . 5 𝑤 ∈ V
5 vex 3477 . . . . 5 𝑧 ∈ V
64, 5elec 8751 . . . 4 (𝑤 ∈ [𝑧] 𝑧 𝑤)
73, 6bitr4di 289 . . 3 ((Prt 𝐴 ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑤 ∈ [𝑧] ))
87eqrdv 2729 . 2 ((Prt 𝐴 ∧ (𝑣𝐴𝑧𝑣)) → 𝑣 = [𝑧] )
98ex 412 1 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wrex 3069   class class class wbr 5148  {copab 5210  [cec 8705  Prt wprt 38045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8709  df-prt 38046
This theorem is referenced by:  prter2  38055
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