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Theorem 0wlkonlem2 29112
Description: Lemma 2 for 0wlkon 29113 and 0trlon 29117. (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 ∈ (𝑉 ↑pm (0...0)))

Proof of Theorem 0wlkonlem2
StepHypRef Expression
1 ovex 7394 . 2 (0...0) ∈ V
2 0wlk.v . . 3 𝑉 = (Vtx‘𝐺)
32fvexi 6860 . 2 𝑉 ∈ V
4 simpl 484 . 2 ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃:(0...0)⟶𝑉)
5 fpmg 8812 . 2 (((0...0) ∈ V ∧ 𝑉 ∈ V ∧ 𝑃:(0...0)⟶𝑉) → 𝑃 ∈ (𝑉 ↑pm (0...0)))
61, 3, 4, 5mp3an12i 1466 1 ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉 ↑pm (0...0)))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447  âŸ¶wf 6496  â€˜cfv 6500  (class class class)co 7361   ↑pm cpm 8772  0cc0 11059  ...cfz 13433  Vtxcvtx 27996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-pm 8774
This theorem is referenced by:  0wlkon  29113  0trlon  29117  0pthon  29120
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