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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 1145 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph)) | |
2 | 0ss 4392 | . . . 4 ⊢ ∅ ⊆ (Vtx‘𝐺) | |
3 | sseq1 3998 | . . . 4 ⊢ ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺))) | |
4 | 2, 3 | mpbiri 257 | . . 3 ⊢ ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) |
5 | 4 | 3ad2ant3 1132 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) |
6 | eqid 2725 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
7 | 6 | uhgrfun 28923 | . . 3 ⊢ (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆)) |
8 | 7 | 3ad2ant2 1131 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆)) |
9 | edgval 28906 | . . 3 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆) | |
10 | uhgr0vb 28929 | . . . . . . . 8 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅)) | |
11 | rneq 5932 | . . . . . . . . 9 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅) | |
12 | rn0 5922 | . . . . . . . . 9 ⊢ ran ∅ = ∅ | |
13 | 11, 12 | eqtrdi 2781 | . . . . . . . 8 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅) |
14 | 10, 13 | biimtrdi 252 | . . . . . . 7 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅)) |
15 | 14 | ex 411 | . . . . . 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 | eqtrid 2777 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅) |
20 | egrsubgr 29134 | . 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3939 ∅c0 4318 class class class wbr 5143 ran crn 5673 Fun wfun 6537 ‘cfv 6543 Vtxcvtx 28853 iEdgciedg 28854 Edgcedg 28904 UHGraphcuhgr 28913 SubGraph csubgr 29124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-edg 28905 df-uhgr 28915 df-subgr 29125 |
This theorem is referenced by: (None) |
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