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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 4007 | . . 3 ⊢ (𝐺 ∈ UHGraph → 𝑉 ⊆ 𝑉) | |
| 2 | isubgriedg.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 3 | isubgriedg.e | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 4 | 2, 3 | isisubgr 47848 | . . 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 29083 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
| 7 | funrel 6583 | . . . . 5 ⊢ (Fun 𝐸 → Rel 𝐸) | |
| 8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Rel 𝐸) |
| 9 | 2, 3 | uhgrf 29079 | . . . . 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 4131 | . . . . . . . . 9 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ∈ 𝒫 𝑉) | |
| 12 | 11 | elpwid 4609 | . . . . . . . 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 3448 | . . . . . 6 ⊢ (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉} = dom 𝐸) |
| 15 | 14 | eqimsscd 4041 | . . . . 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 6040 | . . . 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 4880 | . 2 ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})〉 = 〈𝑉, 𝐸〉) |
| 20 | 5, 19 | eqtrd 2777 | 1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = 〈𝑉, 𝐸〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 〈cop 4632 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 ISubGr cisubgr 47846 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-uhgr 29075 df-isubgr 47847 |
| This theorem is referenced by: (None) |
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