Theorem isubgrupgr 47861

Description: An induced subgraph of a pseudograph is a pseudograph. (Contributed by AV, 14-May-2025.)

Hypothesis

Ref	Expression
isubgrupgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
isubgrupgr	⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UPGraph)

Proof of Theorem isubgrupgr

Step	Hyp	Ref	Expression
1		upgruhgr 29120	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2		isubgrupgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	isubgrsubgr 47860	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)
4	1, 3	sylan 580	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)
5		subupgr 29305	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐺 ISubGr 𝑆) SubGraph 𝐺) → (𝐺 ISubGr 𝑆) ∈ UPGraph)
6	4, 5	syldan 591	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ⊆ wss 3950   class class class wbr 5142  ‘cfv 6560  (class class class)co 7432  Vtxcvtx 29014  UHGraphcuhgr 29074  UPGraphcupgr 29098   SubGraph csubgr 29285   ISubGr cisubgr 47851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-vtx 29016  df-iedg 29017  df-edg 29066  df-uhgr 29076  df-upgr 29100  df-subgr 29286  df-isubgr 47852

This theorem is referenced by: (None)