Theorem upgrun 29150

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 29118	. . . . 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 2735	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8		upgrun.f	. . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻)
9	7, 8	upgrf 29118	. . . . . 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 2741	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻))
13	12	pweqd 4622	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
14	13	difeq1d 4135	. . . . . . 7 ⊢ (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
15	14	rabeqdv 3449	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
16	15	feq3d 6724	. . . . 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 6773	. . 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 5919	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹))
22		dmun 5924	. . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹)
23	21, 22	eqtrdi 2791	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
24		upgrun.v	. . . . . . 7 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
25	24	pweqd 4622	. . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
26	25	difeq1d 4135	. . . . 5 ⊢ (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
27	26	rabeqdv 3449	. . . 4 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
28	20, 23, 27	feq123d 6726	. . 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 2735	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
32		eqid 2735	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
33	31, 32	isupgr 29116	. . 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 1537   ∈ wcel 2106  {crab 3433   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962  ∅c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  dom cdm 5689  ⟶wf 6559  ‘cfv 6563   ≤ cle 11294  2c2 12319  ♯chash 14366  Vtxcvtx 29028  iEdgciedg 29029  UPGraphcupgr 29112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-upgr 29114

This theorem is referenced by:  upgrunop  29151  uspgrun  29220