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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 3624 | . . 3 ⊢ (Vtx‘𝐺) ⊆ (Vtx‘𝐺) | |
2 | ssid 3624 | . . 3 ⊢ (iEdg‘𝐺) ⊆ (iEdg‘𝐺) | |
3 | 1, 2 | pm3.2i 471 | . 2 ⊢ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)) |
4 | eqid 2622 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 4 | uhgrfun 25961 | . . 3 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
6 | id 22 | . . 3 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph ) | |
7 | eqid 2622 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | 7, 7, 4, 4 | uhgrissubgr 26167 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph ) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)))) |
9 | 5, 6, 8 | mpd3an23 1426 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)))) |
10 | 3, 9 | mpbiri 248 | 1 ⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 Fun wfun 5882 ‘cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 UHGraph cuhgr 25951 SubGraph csubgr 26159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-edg 25940 df-uhgr 25953 df-subgr 26160 |
This theorem is referenced by: (None) |
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