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| Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgrsubgr | Structured version Visualization version GIF version | ||
| Description: An induced subgraph of a hypergraph is a subgraph of the hypergraph. (Contributed by AV, 14-May-2025.) | 
| Ref | Expression | 
|---|---|
| isubgrvtx.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| Ref | Expression | 
|---|---|
| isubgrsubgr | ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isubgrvtx.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | 1 | isubgrvtx 47858 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆) | 
| 3 | simpr 484 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉) | |
| 4 | 2, 3 | eqsstrd 4017 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉) | 
| 5 | eqid 2736 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 6 | 1, 5 | isubgriedg 47854 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) | 
| 7 | resss 6018 | . . 3 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺) | |
| 8 | 6, 7 | eqsstrdi 4027 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺)) | 
| 9 | simpl 482 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝐺 ∈ UHGraph) | |
| 10 | 5 | uhgrfun 29084 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) | 
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → Fun (iEdg‘𝐺)) | 
| 12 | 1 | isubgruhgr 47859 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph) | 
| 13 | eqid 2736 | . . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑆)) = (Vtx‘(𝐺 ISubGr 𝑆)) | |
| 14 | eqid 2736 | . . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑆)) = (iEdg‘(𝐺 ISubGr 𝑆)) | |
| 15 | 13, 1, 14, 5 | uhgrissubgr 29293 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ (𝐺 ISubGr 𝑆) ∈ UHGraph) → ((𝐺 ISubGr 𝑆) SubGraph 𝐺 ↔ ((Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉 ∧ (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺)))) | 
| 16 | 9, 11, 12, 15 | syl3anc 1372 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((𝐺 ISubGr 𝑆) SubGraph 𝐺 ↔ ((Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉 ∧ (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺)))) | 
| 17 | 4, 8, 16 | mpbir2and 713 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 class class class wbr 5142 dom cdm 5684 ↾ cres 5686 Fun wfun 6554 ‘cfv 6560 (class class class)co 7432 Vtxcvtx 29014 iEdgciedg 29015 UHGraphcuhgr 29074 SubGraph csubgr 29285 ISubGr cisubgr 47851 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-vtx 29016 df-iedg 29017 df-edg 29066 df-uhgr 29076 df-subgr 29286 df-isubgr 47852 | 
| This theorem is referenced by: isubgrupgr 47861 isubgrumgr 47862 isubgrusgr 47863 | 
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