Theorem upgrss 29119

Description: An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 29-Nov-2020.)

Hypotheses

Ref	Expression
isupgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isupgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
upgrss	⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉)

Proof of Theorem upgrss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4089	. . . 4 ⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅})
2		difss 4145	. . . 4 ⊢ (𝒫 𝑉 ∖ {∅}) ⊆ 𝒫 𝑉
3	1, 2	sstri 4004	. . 3 ⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ 𝒫 𝑉
4		isupgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5		isupgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
6	4, 5	upgrf 29117	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
7	6	ffvelcdmda 7103	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
8	3, 7	sselid 3992	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ 𝒫 𝑉)
9	8	elpwid 4613	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  {crab 3432   ∖ cdif 3959   ⊆ wss 3962  ∅c0 4338  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  dom cdm 5688  ‘cfv 6562   ≤ cle 11293  2c2 12318  ♯chash 14365  Vtxcvtx 29027  iEdgciedg 29028  UPGraphcupgr 29111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-upgr 29113

This theorem is referenced by:  upgrex  29123  isuspgrim0  47809