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Theorem 0wlkonlem2 28016
 Description: Lemma 2 for 0wlkon 28017 and 0trlon 28021. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Hypothesis
Ref Expression
0wlk.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
0wlkonlem2 ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉pm (0...0)))

Proof of Theorem 0wlkonlem2
StepHypRef Expression
1 ovex 7189 . 2 (0...0) ∈ V
2 0wlk.v . . 3 𝑉 = (Vtx‘𝐺)
32fvexi 6677 . 2 𝑉 ∈ V
4 simpl 486 . 2 ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃:(0...0)⟶𝑉)
5 fpmg 8463 . 2 (((0...0) ∈ V ∧ 𝑉 ∈ V ∧ 𝑃:(0...0)⟶𝑉) → 𝑃 ∈ (𝑉pm (0...0)))
61, 3, 4, 5mp3an12i 1462 1 ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉pm (0...0)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ↑pm cpm 8423  0cc0 10588  ...cfz 12952  Vtxcvtx 26901 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-pm 8425 This theorem is referenced by:  0wlkon  28017  0trlon  28021  0pthon  28024
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