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Theorem isubgrsubgr 47793
Description: An induced subgraph of a hypergraph is a subgraph of the hypergraph. (Contributed by AV, 14-May-2025.)
Hypothesis
Ref Expression
isubgrvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
isubgrsubgr ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)

Proof of Theorem isubgrsubgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isubgrvtx.v . . . 4 𝑉 = (Vtx‘𝐺)
21isubgrvtx 47791 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
3 simpr 484 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → 𝑆𝑉)
42, 3eqsstrd 4034 . 2 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉)
5 eqid 2735 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
61, 5isubgriedg 47787 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (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)
86, 7eqsstrdi 4050 . 2 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺))
9 simpl 482 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → 𝐺 ∈ UHGraph)
105uhgrfun 29098 . . . 4 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
1110adantr 480 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → Fun (iEdg‘𝐺))
121isubgruhgr 47792 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (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))
1513, 1, 14, 5uhgrissubgr 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))))
169, 11, 12, 15syl3anc 1370 . 2 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → ((𝐺 ISubGr 𝑆) SubGraph 𝐺 ↔ ((Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉 ∧ (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺))))
174, 8, 16mpbir2and 713 1 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (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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