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Mirrors > Home > MPE Home > Th. List > uhgrun | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
uhgrun.g | ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
uhgrun.h | ⊢ (𝜑 → 𝐻 ∈ 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 | ⊢ (𝜑 → 𝑈 ∈ 𝑊) |
uhgrun.v | ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) |
uhgrun.un | ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) |
Ref | Expression |
---|---|
uhgrun | ⊢ (𝜑 → 𝑈 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrun.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ UHGraph) | |
2 | uhgrun.vg | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | uhgrun.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | 2, 3 | uhgrf 28322 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
6 | uhgrun.h | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ UHGraph) | |
7 | eqid 2733 | . . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻) | |
8 | uhgrun.f | . . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻) | |
9 | 7, 8 | uhgrf 28322 | . . . . . 6 ⊢ (𝐻 ∈ UHGraph → 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅})) |
10 | 6, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅})) |
11 | uhgrun.vh | . . . . . . . . 9 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
12 | 11 | eqcomd 2739 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻)) |
13 | 12 | pweqd 4620 | . . . . . . 7 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻)) |
14 | 13 | difeq1d 4122 | . . . . . 6 ⊢ (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅})) |
15 | 14 | feq3d 6705 | . . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))) |
16 | 10, 15 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅})) |
17 | uhgrun.i | . . . 4 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
18 | 5, 16, 17 | fun2d 6756 | . . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅})) |
19 | uhgrun.un | . . . 4 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) | |
20 | 19 | dmeqd 5906 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹)) |
21 | dmun 5911 | . . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹) | |
22 | 20, 21 | eqtrdi 2789 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹)) |
23 | uhgrun.v | . . . . . 6 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | |
24 | 23 | pweqd 4620 | . . . . 5 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉) |
25 | 24 | difeq1d 4122 | . . . 4 ⊢ (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
26 | 19, 22, 25 | feq123d 6707 | . . 3 ⊢ (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}) ↔ (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅}))) |
27 | 18, 26 | mpbird 257 | . 2 ⊢ (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})) |
28 | uhgrun.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊) | |
29 | eqid 2733 | . . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈) | |
30 | eqid 2733 | . . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈) | |
31 | 29, 30 | isuhgr 28320 | . . 3 ⊢ (𝑈 ∈ 𝑊 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}))) |
32 | 28, 31 | syl 17 | . 2 ⊢ (𝜑 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}))) |
33 | 27, 32 | mpbird 257 | 1 ⊢ (𝜑 → 𝑈 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ∅c0 4323 𝒫 cpw 4603 {csn 4629 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 Vtxcvtx 28256 iEdgciedg 28257 UHGraphcuhgr 28316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-uhgr 28318 |
This theorem is referenced by: uhgrunop 28335 ushgrun 28336 |
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