Theorem isubgrsubgr 47849

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 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)

Proof of Theorem isubgrsubgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isubgrvtx.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isubgrvtx 47847	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
3		simpr 484	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
4	2, 3	eqsstrd 3998	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉)
5		eqid 2736	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	1, 5	isubgriedg 47843	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
7		resss 5993	. . 3 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺)
8	6, 7	eqsstrdi 4008	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺))
9		simpl 482	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝐺 ∈ UHGraph)
10	5	uhgrfun 29050	. . . 4 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
11	10	adantr 480	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → Fun (iEdg‘𝐺))
12	1	isubgruhgr 47848	. . 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 29259	. . 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 1373	. 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)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3420   ⊆ wss 3931   class class class wbr 5124  dom cdm 5659   ↾ cres 5661  Fun wfun 6530  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 28980  iEdgciedg 28981  UHGraphcuhgr 29040   SubGraph csubgr 29251   ISubGr cisubgr 47840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-vtx 28982  df-iedg 28983  df-edg 29032  df-uhgr 29042  df-subgr 29252  df-isubgr 47841

This theorem is referenced by:  isubgrupgr  47850  isubgrumgr  47851  isubgrusgr  47852