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Mirrors > Home > MPE Home > Th. List > uhgrspan | Structured version Visualization version GIF version |
Description: A spanning subgraph 𝑆 of a hypergraph 𝐺 is a hypergraph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.) |
Ref | Expression |
---|---|
uhgrspan.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrspan.e | ⊢ 𝐸 = (iEdg‘𝐺) |
uhgrspan.s | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
uhgrspan.q | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
uhgrspan.r | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) |
uhgrspan.g | ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
Ref | Expression |
---|---|
uhgrspan | ⊢ (𝜑 → 𝑆 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrspan.g | . 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph) | |
2 | uhgrspan.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | uhgrspan.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | uhgrspan.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
5 | uhgrspan.q | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
6 | uhgrspan.r | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) | |
7 | 2, 3, 4, 5, 6, 1 | uhgrspansubgr 27000 | . 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
8 | subuhgr 26995 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph) | |
9 | 1, 7, 8 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑆 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ↾ cres 5550 ‘cfv 6348 Vtxcvtx 26708 iEdgciedg 26709 UHGraphcuhgr 26768 SubGraph csubgr 26976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-edg 26760 df-uhgr 26770 df-subgr 26977 |
This theorem is referenced by: uhgrspanop 27005 |
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