Theorem nbgrssvtx 28579

Description: The neighbors of a vertex 𝐾 in a graph form a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.)

Hypothesis

Ref	Expression
nbgrisvtx.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbgrssvtx	⊢ (𝐺 NeighbVtx 𝐾) ⊆ 𝑉

Proof of Theorem nbgrssvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgrisvtx.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrisvtx 28578	. 2 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝐾) → 𝑛 ∈ 𝑉)
3	2	ssriv 3985	1 ⊢ (𝐺 NeighbVtx 𝐾) ⊆ 𝑉

Syntax hints:   = wceq 1542   ⊆ wss 3947  ‘cfv 6540  (class class class)co 7404  Vtxcvtx 28236   NeighbVtx cnbgr 28569

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 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  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-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-nbgr 28570

This theorem is referenced by:  fusgreghash2wspv  29568