Theorem nbgrssvtx 29923

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 29922	. 2 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝐾) → 𝑛 ∈ 𝑉)
3	2	ssriv 3912	1 ⊢ (𝐺 NeighbVtx 𝐾) ⊆ 𝑉

This proof depends on syntax axioms:   = wceq 1569   ⊆ wss 3895  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29574   NeighbVtx cnbgr 29913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-nbgr 29914

This theorem is used by:  fusgreghash2wspv  30936