Theorem nbgrclOLD 26425

Description: Obsolete version of nbgrcl 26424 as of 12-Feb-2022. (Contributed by AV, 6-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nbgrclOLD	⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))

Proof of Theorem nbgrclOLD

Dummy variables 𝑔 𝑒 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nbgr 26422	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
2	1	mpt2xeldm 7504	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ V ∧ 𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
3		csbfv 6392	. . . 4 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
4	3	eleq2i 2829	. . 3 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) ↔ 𝑋 ∈ (Vtx‘𝐺))
5	4	biimpi 206	. 2 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → 𝑋 ∈ (Vtx‘𝐺))
6	2, 5	simpl2im 659	1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))

Syntax hints:   → wi 4   ∈ wcel 2137  ∃wrex 3049  {crab 3052  Vcvv 3338  ⦋csb 3672   ∖ cdif 3710   ⊆ wss 3713  {csn 4319  {cpr 4321  ‘cfv 6047  (class class class)co 6811  Vtxcvtx 26071  Edgcedg 26136   NeighbVtx cnbgr 26421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-1st 7331  df-2nd 7332  df-nbgr 26422

This theorem is referenced by:  nbgrelOLD  26431