Theorem upgrfn 29014

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 29013	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		fndm 6621	. . . 4 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
5	4	feq2d 6672	. . 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 1540   ∈ wcel 2109  {crab 3405   ∖ cdif 3911  ∅c0 4296  𝒫 cpw 4563  {csn 4589   class class class wbr 5107  dom cdm 5638   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511   ≤ cle 11209  2c2 12241  ♯chash 14295  Vtxcvtx 28923  iEdgciedg 28924  UPGraphcupgr 29007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-upgr 29009

This theorem is referenced by:  upgrn0  29016  upgrle  29017