![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0uhgrsubgr | Structured version Visualization version GIF version |
Description: The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.) |
Ref | Expression |
---|---|
0uhgrsubgr | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 1145 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph)) | |
2 | 0ss 4304 | . . . 4 ⊢ ∅ ⊆ (Vtx‘𝐺) | |
3 | sseq1 3940 | . . . 4 ⊢ ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺))) | |
4 | 2, 3 | mpbiri 261 | . . 3 ⊢ ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) |
5 | 4 | 3ad2ant3 1132 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) |
6 | eqid 2798 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
7 | 6 | uhgrfun 26859 | . . 3 ⊢ (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆)) |
8 | 7 | 3ad2ant2 1131 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆)) |
9 | edgval 26842 | . . 3 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆) | |
10 | uhgr0vb 26865 | . . . . . . . 8 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅)) | |
11 | rneq 5770 | . . . . . . . . 9 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅) | |
12 | rn0 5760 | . . . . . . . . 9 ⊢ ran ∅ = ∅ | |
13 | 11, 12 | eqtrdi 2849 | . . . . . . . 8 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅) |
14 | 10, 13 | syl6bi 256 | . . . . . . 7 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅)) |
15 | 14 | ex 416 | . . . . . 6 ⊢ (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅))) |
16 | 15 | pm2.43a 54 | . . . . 5 ⊢ (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)) |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅))) |
18 | 17 | 3imp 1108 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → ran (iEdg‘𝑆) = ∅) |
19 | 9, 18 | syl5eq 2845 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅) |
20 | egrsubgr 27067 | . 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺) | |
21 | 1, 5, 8, 19, 20 | syl112anc 1371 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ran crn 5520 Fun wfun 6318 ‘cfv 6324 Vtxcvtx 26789 iEdgciedg 26790 Edgcedg 26840 UHGraphcuhgr 26849 SubGraph csubgr 27057 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-edg 26841 df-uhgr 26851 df-subgr 27058 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |