Theorem isubgrsubgr 47994

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 47992	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
3		simpr 484	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
4	2, 3	eqsstrd 3965	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉)
5		eqid 2733	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	1, 5	isubgriedg 47988	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
7		resss 5954	. . 3 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺)
8	6, 7	eqsstrdi 3975	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺))
9		simpl 482	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝐺 ∈ UHGraph)
10	5	uhgrfun 29046	. . . 4 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
11	10	adantr 480	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → Fun (iEdg‘𝐺))
12	1	isubgruhgr 47993	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph)
13		eqid 2733	. . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑆)) = (Vtx‘(𝐺 ISubGr 𝑆))
14		eqid 2733	. . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑆)) = (iEdg‘(𝐺 ISubGr 𝑆))
15	13, 1, 14, 5	uhgrissubgr 29255	. . 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 1541   ∈ wcel 2113  {crab 3396   ⊆ wss 3898   class class class wbr 5093  dom cdm 5619   ↾ cres 5621  Fun wfun 6480  ‘cfv 6486  (class class class)co 7352  Vtxcvtx 28976  iEdgciedg 28977  UHGraphcuhgr 29036   SubGraph csubgr 29247   ISubGr cisubgr 47985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-vtx 28978  df-iedg 28979  df-edg 29028  df-uhgr 29038  df-subgr 29248  df-isubgr 47986

This theorem is referenced by:  isubgrupgr  47995  isubgrumgr  47996  isubgrusgr  47997