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Theorem prtlem19 38860
Description: Lemma for prter2 38863. (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 38859 . . . . 5 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
32imp 406 . . . 4 ((Prt 𝐴 ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑧 𝑤))
4 vex 3482 . . . . 5 𝑤 ∈ V
5 vex 3482 . . . . 5 𝑧 ∈ V
64, 5elec 8790 . . . 4 (𝑤 ∈ [𝑧] 𝑧 𝑤)
73, 6bitr4di 289 . . 3 ((Prt 𝐴 ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑤 ∈ [𝑧] ))
87eqrdv 2733 . 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 206  wa 395   = wceq 1537  wcel 2106  wrex 3068   class class class wbr 5148  {copab 5210  [cec 8742  Prt wprt 38853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-prt 38854
This theorem is referenced by:  prter2  38863
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