Theorem ushgrun 26424

Description: The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

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

Assertion

Ref	Expression
ushgrun	⊢ (𝜑 → 𝑈 ∈ UHGraph)

Proof of Theorem ushgrun

Step	Hyp	Ref	Expression
1		ushgrun.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ USHGraph)
2		ushgruhgr 26417	. . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
4		ushgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph)
5		ushgruhgr 26417	. . 3 ⊢ (𝐻 ∈ USHGraph → 𝐻 ∈ UHGraph)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐻 ∈ UHGraph)
7		ushgrun.e	. 2 ⊢ 𝐸 = (iEdg‘𝐺)
8		ushgrun.f	. 2 ⊢ 𝐹 = (iEdg‘𝐻)
9		ushgrun.vg	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
10		ushgrun.vh	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
11		ushgrun.i	. 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
12		ushgrun.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
13		ushgrun.v	. 2 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
14		ushgrun.un	. 2 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))
15	3, 6, 7, 8, 9, 10, 11, 12, 13, 14	uhgrun 26422	1 ⊢ (𝜑 → 𝑈 ∈ UHGraph)

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107   ∪ cun 3790   ∩ cin 3791  ∅c0 4141  dom cdm 5355  ‘cfv 6135  Vtxcvtx 26344  iEdgciedg 26345  UHGraphcuhgr 26404  USHGraphcushgr 26405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fv 6143  df-uhgr 26406  df-ushgr 26407

This theorem is referenced by: (None)