Theorem uvtxssvtx 29463

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 29462	. 2 ⊢ (𝑛 ∈ (UnivVtx‘𝐺) → 𝑛 ∈ 𝑉)
3	2	ssriv 3937	1 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉

Syntax hints:   = wceq 1541   ⊆ wss 3901  ‘cfv 6492  Vtxcvtx 29069  UnivVtxcuvtx 29458

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-uvtx 29459

This theorem is referenced by:  iscplgr  29488  vdiscusgrb  29604