Theorem upgrfn 29109

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
upgrfn	⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Distinct variable groups:   𝑥,𝐺   𝑥,𝑉

Allowed substitution hints:   𝐴(𝑥)   𝐸(𝑥)

Proof of Theorem upgrfn

Step	Hyp	Ref	Expression
1		isupgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		isupgr.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	upgrf 29108	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		fndm 6593	. . . 4 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
5	4	feq2d 6644	. . 3 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
6	3, 5	syl5ibcom 245	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
7	6	imp 406	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3397   ∖ cdif 3896  ∅c0 4283  𝒫 cpw 4552  {csn 4578   class class class wbr 5096  dom cdm 5622   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490   ≤ cle 11165  2c2 12198  ♯chash 14251  Vtxcvtx 29018  iEdgciedg 29019  UPGraphcupgr 29102

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-ext 2706  ax-nul 5249

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-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-upgr 29104

This theorem is referenced by:  upgrn0  29111  upgrle  29112