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Theorem prtlem19 38051
Description: Lemma for prter2 38054. (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 38050 . . . . 5 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
32imp 405 . . . 4 ((Prt 𝐴 ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑧 𝑤))
4 vex 3476 . . . . 5 𝑤 ∈ V
5 vex 3476 . . . . 5 𝑧 ∈ V
64, 5elec 8749 . . . 4 (𝑤 ∈ [𝑧] 𝑧 𝑤)
73, 6bitr4di 288 . . 3 ((Prt 𝐴 ∧ (𝑣𝐴𝑧𝑣)) → (𝑤𝑣𝑤 ∈ [𝑧] ))
87eqrdv 2728 . 2 ((Prt 𝐴 ∧ (𝑣𝐴𝑧𝑣)) → 𝑣 = [𝑧] )
98ex 411 1 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wrex 3068   class class class wbr 5147  {copab 5209  [cec 8703  Prt wprt 38044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ec 8707  df-prt 38045
This theorem is referenced by:  prter2  38054
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