Theorem upgrun 29052

Description: The union 𝑈 of two pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
upgrun.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)
upgrun.h	⊢ (𝜑 → 𝐻 ∈ UPGraph)
upgrun.e	⊢ 𝐸 = (iEdg‘𝐺)
upgrun.f	⊢ 𝐹 = (iEdg‘𝐻)
upgrun.vg	⊢ 𝑉 = (Vtx‘𝐺)
upgrun.vh	⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
upgrun.i	⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
upgrun.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
upgrun.v	⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
upgrun.un	⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))

Assertion

Ref	Expression
upgrun	⊢ (𝜑 → 𝑈 ∈ UPGraph)

Proof of Theorem upgrun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgrun.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
2		upgrun.vg	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
3		upgrun.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
4	2, 3	upgrf 29020	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
6		upgrun.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ UPGraph)
7		eqid 2730	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8		upgrun.f	. . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻)
9	7, 8	upgrf 29020	. . . . . 6 ⊢ (𝐻 ∈ UPGraph → 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
10	6, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
11		upgrun.vh	. . . . . . . . . 10 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
12	11	eqcomd 2736	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻))
13	12	pweqd 4583	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
14	13	difeq1d 4091	. . . . . . 7 ⊢ (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
15	14	rabeqdv 3424	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
16	15	feq3d 6676	. . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
17	10, 16	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18		upgrun.i	. . . 4 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
19	5, 17, 18	fun2d 6727	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
20		upgrun.un	. . . 4 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))
21	20	dmeqd 5872	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹))
22		dmun 5877	. . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹)
23	21, 22	eqtrdi 2781	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
24		upgrun.v	. . . . . . 7 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
25	24	pweqd 4583	. . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
26	25	difeq1d 4091	. . . . 5 ⊢ (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
27	26	rabeqdv 3424	. . . 4 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
28	20, 23, 27	feq123d 6680	. . 3 ⊢ (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
29	19, 28	mpbird 257	. 2 ⊢ (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
30		upgrun.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
31		eqid 2730	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
32		eqid 2730	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
33	31, 32	isupgr 29018	. . 3 ⊢ (𝑈 ∈ 𝑊 → (𝑈 ∈ UPGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
34	30, 33	syl 17	. 2 ⊢ (𝜑 → (𝑈 ∈ UPGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
35	29, 34	mpbird 257	1 ⊢ (𝜑 → 𝑈 ∈ UPGraph)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {crab 3408   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916  ∅c0 4299  𝒫 cpw 4566  {csn 4592   class class class wbr 5110  dom cdm 5641  ⟶wf 6510  ‘cfv 6514   ≤ cle 11216  2c2 12248  ♯chash 14302  Vtxcvtx 28930  iEdgciedg 28931  UPGraphcupgr 29014

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-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-upgr 29016

This theorem is referenced by:  upgrunop  29053  uspgrun  29122