Theorem upgrun 29046

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 29014	. . . . 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 2725	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8		upgrun.f	. . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻)
9	7, 8	upgrf 29014	. . . . . 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 2731	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻))
13	12	pweqd 4623	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
14	13	difeq1d 4119	. . . . . . 7 ⊢ (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
15	14	rabeqdv 3434	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
16	15	feq3d 6714	. . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
17	10, 16	mpbird 256	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18		upgrun.i	. . . 4 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
19	5, 17, 18	fun2d 6765	. . 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 5911	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹))
22		dmun 5916	. . . . 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 4623	. . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
26	25	difeq1d 4119	. . . . 5 ⊢ (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
27	26	rabeqdv 3434	. . . 4 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
28	20, 23, 27	feq123d 6716	. . 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 256	. 2 ⊢ (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
30		upgrun.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
31		eqid 2725	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
32		eqid 2725	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
33	31, 32	isupgr 29012	. . 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 256	1 ⊢ (𝜑 → 𝑈 ∈ UPGraph)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {crab 3418   ∖ cdif 3943   ∪ cun 3944   ∩ cin 3945  ∅c0 4324  𝒫 cpw 4606  {csn 4632   class class class wbr 5152  dom cdm 5681  ⟶wf 6549  ‘cfv 6553   ≤ cle 11295  2c2 12314  ♯chash 14342  Vtxcvtx 28924  iEdgciedg 28925  UPGraphcupgr 29008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-upgr 29010

This theorem is referenced by:  upgrunop  29047  uspgrun  29116