Theorem upgrfn 29177

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 29176	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		fndm 6591	. . . 4 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
5	4	feq2d 6642	. . 3 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
6	3, 5	syl5ibcom 246	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
7	6	imp 407	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1543   ∈ wcel 2115  {crab 3388   ∖ cdif 3883  ∅c0 4264  𝒫 cpw 4532  {csn 4558   class class class wbr 5075  dom cdm 5621   Fn wfn 6483  ⟶wf 6484  ‘cfv 6488   ≤ cle 11174  2c2 12230  ♯chash 14286  Vtxcvtx 29086  iEdgciedg 29087  UPGraphcupgr 29170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-ext 2708  ax-nul 5231

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-sb 2070  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2932  df-rab 3389  df-v 3430  df-sbc 3727  df-dif 3889  df-un 3891  df-ss 3903  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-upgr 29172

This theorem is referenced by:  upgrn0  29179  upgrle  29180