Theorem uvtxssvtx 29370

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 29369	. 2 ⊢ (𝑛 ∈ (UnivVtx‘𝐺) → 𝑛 ∈ 𝑉)
3	2	ssriv 3934	1 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉

Syntax hints:   = wceq 1541   ⊆ wss 3898  ‘cfv 6486  Vtxcvtx 28976  UnivVtxcuvtx 29365

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-uvtx 29366

This theorem is referenced by:  iscplgr  29395  vdiscusgrb  29511