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Theorem uhgrun 25740
 Description: The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
Hypotheses
Ref Expression
uhgrun.g (𝜑𝐺 ∈ UHGraph )
uhgrun.h (𝜑𝐻 ∈ UHGraph )
uhgrun.e 𝐸 = (iEdg‘𝐺)
uhgrun.f 𝐹 = (iEdg‘𝐻)
uhgrun.vg 𝑉 = (Vtx‘𝐺)
uhgrun.vh (𝜑 → (Vtx‘𝐻) = 𝑉)
uhgrun.i (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
uhgrun.u (𝜑𝑈𝑊)
uhgrun.v (𝜑 → (Vtx‘𝑈) = 𝑉)
uhgrun.un (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
Assertion
Ref Expression
uhgrun (𝜑𝑈 ∈ UHGraph )

Proof of Theorem uhgrun
StepHypRef Expression
1 uhgrun.g . . . . 5 (𝜑𝐺 ∈ UHGraph )
2 uhgrun.vg . . . . . 6 𝑉 = (Vtx‘𝐺)
3 uhgrun.e . . . . . 6 𝐸 = (iEdg‘𝐺)
42, 3uhgrf 25728 . . . . 5 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
51, 4syl 17 . . . 4 (𝜑𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
6 uhgrun.h . . . . . 6 (𝜑𝐻 ∈ UHGraph )
7 eqid 2610 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘𝐻)
8 uhgrun.f . . . . . . 7 𝐹 = (iEdg‘𝐻)
97, 8uhgrf 25728 . . . . . 6 (𝐻 ∈ UHGraph → 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))
106, 9syl 17 . . . . 5 (𝜑𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))
11 uhgrun.vh . . . . . . . . 9 (𝜑 → (Vtx‘𝐻) = 𝑉)
1211eqcomd 2616 . . . . . . . 8 (𝜑𝑉 = (Vtx‘𝐻))
1312pweqd 4113 . . . . . . 7 (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
1413difeq1d 3689 . . . . . 6 (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
1514feq3d 5945 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅})))
1610, 15mpbird 246 . . . 4 (𝜑𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}))
17 uhgrun.i . . . 4 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
185, 16, 17fun2d 5981 . . 3 (𝜑 → (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅}))
19 uhgrun.un . . . 4 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
2019dmeqd 5248 . . . . 5 (𝜑 → dom (iEdg‘𝑈) = dom (𝐸𝐹))
21 dmun 5253 . . . . 5 dom (𝐸𝐹) = (dom 𝐸 ∪ dom 𝐹)
2220, 21syl6eq 2660 . . . 4 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
23 uhgrun.v . . . . . 6 (𝜑 → (Vtx‘𝑈) = 𝑉)
2423pweqd 4113 . . . . 5 (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
2524difeq1d 3689 . . . 4 (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2619, 22, 25feq123d 5947 . . 3 (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}) ↔ (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅})))
2718, 26mpbird 246 . 2 (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}))
28 uhgrun.u . . 3 (𝜑𝑈𝑊)
29 eqid 2610 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
30 eqid 2610 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
3129, 30isuhgr 25726 . . 3 (𝑈𝑊 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})))
3228, 31syl 17 . 2 (𝜑 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})))
3327, 32mpbird 246 1 (𝜑𝑈 ∈ UHGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  {csn 4125  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724 This theorem is referenced by:  uhgrunop  25741  ushgrun  25742
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