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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgrvtxuhgr | Structured version Visualization version GIF version |
Description: The subgraph induced by the full set of vertices of a hypergraph. (Contributed by AV, 12-May-2025.) |
Ref | Expression |
---|---|
isubgriedg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isubgriedg.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
isubgrvtxuhgr | ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = 〈𝑉, 𝐸〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 4019 | . . 3 ⊢ (𝐺 ∈ UHGraph → 𝑉 ⊆ 𝑉) | |
2 | isubgriedg.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | isubgriedg.e | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | 2, 3 | isisubgr 47786 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ⊆ 𝑉) → (𝐺 ISubGr 𝑉) = 〈𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})〉) |
5 | 1, 4 | mpdan 687 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = 〈𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})〉) |
6 | 3 | uhgrfun 29098 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
7 | funrel 6585 | . . . . 5 ⊢ (Fun 𝐸 → Rel 𝐸) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Rel 𝐸) |
9 | 2, 3 | uhgrf 29094 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
10 | ffvelcdm 7101 | . . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅})) | |
11 | eldifi 4141 | . . . . . . . . 9 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ∈ 𝒫 𝑉) | |
12 | 11 | elpwid 4614 | . . . . . . . 8 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ⊆ 𝑉) |
13 | 10, 12 | syl 17 | . . . . . . 7 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ⊆ 𝑉) |
14 | 13 | rabeqcda 3445 | . . . . . 6 ⊢ (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉} = dom 𝐸) |
15 | 14 | eqimsscd 4053 | . . . . 5 ⊢ (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) |
16 | 9, 15 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UHGraph → dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) |
17 | relssres 6042 | . . . 4 ⊢ ((Rel 𝐸 ∧ dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸) | |
18 | 8, 16, 17 | syl2anc 584 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸) |
19 | 18 | opeq2d 4885 | . 2 ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})〉 = 〈𝑉, 𝐸〉) |
20 | 5, 19 | eqtrd 2775 | 1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = 〈𝑉, 𝐸〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 〈cop 4637 dom cdm 5689 ↾ cres 5691 Rel wrel 5694 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 iEdgciedg 29029 UHGraphcuhgr 29088 ISubGr cisubgr 47784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-uhgr 29090 df-isubgr 47785 |
This theorem is referenced by: (None) |
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