Theorem isubgrsubgr 49182

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 49180	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
3		simpr 490	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
4	2, 3	eqsstrd 3947	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) ⊆ 𝑉)
5		eqid 2763	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	1, 5	isubgriedg 49176	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
7		resss 6004	. . 3 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺)
8	6, 7	eqsstrdi 3957	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) ⊆ (iEdg‘𝐺))
9		simpl 488	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝐺 ∈ UHGraph)
10	5	uhgrfun 29677	. . . 4 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
11	10	adantr 486	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → Fun (iEdg‘𝐺))
12	1	isubgruhgr 49181	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph)
13		eqid 2763	. . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑆)) = (Vtx‘(𝐺 ISubGr 𝑆))
14		eqid 2763	. . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑆)) = (iEdg‘(𝐺 ISubGr 𝑆))
15	13, 1, 14, 5	uhgrissubgr 29889	. . 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 1399	. 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 726	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416   ⊆ wss 3900   class class class wbr 5106  dom cdm 5656   ↾ cres 5658  Fun wfun 6555  ‘cfv 6561  (class class class)co 7446  Vtxcvtx 29607  iEdgciedg 29608  UHGraphcuhgr 29667   SubGraph csubgr 29881   ISubGr cisubgr 49173

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-vtx 29609  df-iedg 29610  df-edg 29659  df-uhgr 29669  df-subgr 29882  df-isubgr 49174

This theorem is used by:  isubgrupgr  49183  isubgrumgr  49184  isubgrusgr  49185