Theorem uvtxssvtx 29777

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 29776	. 2 ⊢ (𝑛 ∈ (UnivVtx‘𝐺) → 𝑛 ∈ 𝑉)
3	2	ssriv 3944	1 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3908  ‘cfv 6543  Vtxcvtx 29383  UnivVtxcuvtx 29772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-uvtx 29773

This theorem is used by:  iscplgr  29802  vdiscusgrb  29917