![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3964 | . . 3 ⊢ (Vtx‘𝐺) ⊆ (Vtx‘𝐺) | |
2 | ssid 3964 | . . 3 ⊢ (iEdg‘𝐺) ⊆ (iEdg‘𝐺) | |
3 | 1, 2 | pm3.2i 471 | . 2 ⊢ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)) |
4 | eqid 2736 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 4 | uhgrfun 27903 | . . 3 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
6 | id 22 | . . 3 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph) | |
7 | eqid 2736 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | 7, 7, 4, 4 | uhgrissubgr 28109 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)))) |
9 | 5, 6, 8 | mpd3an23 1463 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)))) |
10 | 3, 9 | mpbiri 257 | 1 ⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ⊆ wss 3908 class class class wbr 5103 Fun wfun 6487 ‘cfv 6493 Vtxcvtx 27833 iEdgciedg 27834 UHGraphcuhgr 27893 SubGraph csubgr 28101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-edg 27885 df-uhgr 27895 df-subgr 28102 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |