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Mathbox for Alexander van der Vekens |
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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 47791 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆) |
3 | simpr 484 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉) | |
4 | 2, 3 | eqsstrd 4034 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉) |
5 | eqid 2735 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
6 | 1, 5 | isubgriedg 47787 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) |
7 | resss 6022 | . . 3 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺) | |
8 | 6, 7 | eqsstrdi 4050 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺)) |
9 | simpl 482 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝐺 ∈ UHGraph) | |
10 | 5 | uhgrfun 29098 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → Fun (iEdg‘𝐺)) |
12 | 1 | isubgruhgr 47792 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph) |
13 | eqid 2735 | . . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑆)) = (Vtx‘(𝐺 ISubGr 𝑆)) | |
14 | eqid 2735 | . . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑆)) = (iEdg‘(𝐺 ISubGr 𝑆)) | |
15 | 13, 1, 14, 5 | uhgrissubgr 29307 | . . 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 1370 | . 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 1537 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 class class class wbr 5148 dom cdm 5689 ↾ cres 5691 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 iEdgciedg 29029 UHGraphcuhgr 29088 SubGraph csubgr 29299 ISubGr cisubgr 47784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-vtx 29030 df-iedg 29031 df-edg 29080 df-uhgr 29090 df-subgr 29300 df-isubgr 47785 |
This theorem is referenced by: isubgrupgr 47794 isubgrumgr 47795 isubgrusgr 47796 |
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