Theorem nbgrssvtx 28867

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 28866	. 2 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝐾) → 𝑛 ∈ 𝑉)
3	2	ssriv 3986	1 ⊢ (𝐺 NeighbVtx 𝐾) ⊆ 𝑉

Syntax hints:   = wceq 1540   ⊆ wss 3948  ‘cfv 6543  (class class class)co 7412  Vtxcvtx 28524   NeighbVtx cnbgr 28857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-nbgr 28858

This theorem is referenced by:  fusgreghash2wspv  29856