Theorem uspgrun 29206

Description: The union 𝑈 of two simple 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, 16-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
uspgrun	⊢ (𝜑 → 𝑈 ∈ UPGraph)

Proof of Theorem uspgrun

Step	Hyp	Ref	Expression
1		uspgrun.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ USPGraph)
2		uspgrupgr 29196	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
4		uspgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph)
5		uspgrupgr 29196	. . 3 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐻 ∈ UPGraph)
7		uspgrun.e	. 2 ⊢ 𝐸 = (iEdg‘𝐺)
8		uspgrun.f	. 2 ⊢ 𝐹 = (iEdg‘𝐻)
9		uspgrun.vg	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
10		uspgrun.vh	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
11		uspgrun.i	. 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
12		uspgrun.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
13		uspgrun.v	. 2 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
14		uspgrun.un	. 2 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))
15	3, 6, 7, 8, 9, 10, 11, 12, 13, 14	upgrun 29136	1 ⊢ (𝜑 → 𝑈 ∈ UPGraph)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ∪ cun 3948   ∩ cin 3949  ∅c0 4332  dom cdm 5684  ‘cfv 6560  Vtxcvtx 29014  iEdgciedg 29015  UPGraphcupgr 29098  USPGraphcuspgr 29166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fv 6568  df-upgr 29100  df-uspgr 29168

This theorem is referenced by: (None)