Theorem uvtxssvtx 27172

Description: The set of the universal vertices is a subset of the set of all vertices. (Contributed by AV, 23-Dec-2020.)

Hypothesis

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxssvtx	⊢ (UnivVtx‘𝐺) ⊆ 𝑉

Proof of Theorem uvtxssvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	uvtxisvtx 27171	. 2 ⊢ (𝑛 ∈ (UnivVtx‘𝐺) → 𝑛 ∈ 𝑉)
3	2	ssriv 3971	1 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉

Syntax hints:   = wceq 1537   ⊆ wss 3936  ‘cfv 6355  Vtxcvtx 26781  UnivVtxcuvtx 27167

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-uvtx 27168

This theorem is referenced by:  iscplgr  27197  vdiscusgrb  27312