Theorem upgrss 29161

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 4032	. . . 4 ⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅})
2		difss 4088	. . . 4 ⊢ (𝒫 𝑉 ∖ {∅}) ⊆ 𝒫 𝑉
3	1, 2	sstri 3943	. . 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 29159	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
7	6	ffvelcdmda 7029	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
8	3, 7	sselid 3931	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ 𝒫 𝑉)
9	8	elpwid 4563	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580   class class class wbr 5098  dom cdm 5624  ‘cfv 6492   ≤ cle 11167  2c2 12200  ♯chash 14253  Vtxcvtx 29069  iEdgciedg 29070  UPGraphcupgr 29153

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-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-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  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-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-upgr 29155

This theorem is referenced by:  upgrex  29165  isuspgrim0  48140