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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 1144 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph)) | |
2 | 0ss 4350 | . . . 4 ⊢ ∅ ⊆ (Vtx‘𝐺) | |
3 | sseq1 3992 | . . . 4 ⊢ ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺))) | |
4 | 2, 3 | mpbiri 260 | . . 3 ⊢ ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) |
5 | 4 | 3ad2ant3 1131 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) |
6 | eqid 2821 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
7 | 6 | uhgrfun 26851 | . . 3 ⊢ (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆)) |
8 | 7 | 3ad2ant2 1130 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆)) |
9 | edgval 26834 | . . 3 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆) | |
10 | uhgr0vb 26857 | . . . . . . . 8 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅)) | |
11 | rneq 5806 | . . . . . . . . 9 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅) | |
12 | rn0 5796 | . . . . . . . . 9 ⊢ ran ∅ = ∅ | |
13 | 11, 12 | syl6eq 2872 | . . . . . . . 8 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅) |
14 | 10, 13 | syl6bi 255 | . . . . . . 7 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅)) |
15 | 14 | ex 415 | . . . . . 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 1107 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → ran (iEdg‘𝑆) = ∅) |
19 | 9, 18 | syl5eq 2868 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅) |
20 | egrsubgr 27059 | . 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺) | |
21 | 1, 5, 8, 19, 20 | syl112anc 1370 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 ran crn 5556 Fun wfun 6349 ‘cfv 6355 Vtxcvtx 26781 iEdgciedg 26782 Edgcedg 26832 UHGraphcuhgr 26841 SubGraph csubgr 27049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-edg 26833 df-uhgr 26843 df-subgr 27050 |
This theorem is referenced by: (None) |
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