Theorem 0wlkonlem2 30189

Description: Lemma 2 for 0wlkon 30190 and 0trlon 30194. (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

Step	Hyp	Ref	Expression
1		ovex 7400	. 2 ⊢ (0...0) ∈ V
2		0wlk.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	fvexi 6854	. 2 ⊢ 𝑉 ∈ V
4		simpl 482	. 2 ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃:(0...0)⟶𝑉)
5		fpmg 8816	. 2 ⊢ (((0...0) ∈ V ∧ 𝑉 ∈ V ∧ 𝑃:(0...0)⟶𝑉) → 𝑃 ∈ (𝑉 ↑pm (0...0)))
6	1, 3, 4, 5	mp3an12i 1468	1 ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉 ↑pm (0...0)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_pm cpm 8774  0cc0 11038  ...cfz 13461  Vtxcvtx 29065

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-pm 8776

This theorem is referenced by:  0wlkon  30190  0trlon  30194  0pthon  30197