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Mirrors > Home > MPE Home > Th. List > uhgrsubgrself | Structured version Visualization version GIF version |
Description: A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.) |
Ref | Expression |
---|---|
uhgrsubgrself | ⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3986 | . . 3 ⊢ (Vtx‘𝐺) ⊆ (Vtx‘𝐺) | |
2 | ssid 3986 | . . 3 ⊢ (iEdg‘𝐺) ⊆ (iEdg‘𝐺) | |
3 | 1, 2 | pm3.2i 471 | . 2 ⊢ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)) |
4 | eqid 2818 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 4 | uhgrfun 26778 | . . 3 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
6 | id 22 | . . 3 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph) | |
7 | eqid 2818 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | 7, 7, 4, 4 | uhgrissubgr 26984 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)))) |
9 | 5, 6, 8 | mpd3an23 1454 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)))) |
10 | 3, 9 | mpbiri 259 | 1 ⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 class class class wbr 5057 Fun wfun 6342 ‘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-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: (None) |
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