Theorem isubgrupgr 48968

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 29680	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2		isubgrupgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	isubgrsubgr 48967	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)
4	1, 3	sylan 592	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)
5		subupgr 29868	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐺 ISubGr 𝑆) SubGraph 𝐺) → (𝐺 ISubGr 𝑆) ∈ UPGraph)
6	4, 5	syldan 603	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29574  UHGraphcuhgr 29634  UPGraphcupgr 29658   SubGraph csubgr 29848   ISubGr cisubgr 48958

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-vtx 29576  df-iedg 29577  df-edg 29626  df-uhgr 29636  df-upgr 29660  df-subgr 29849  df-isubgr 48959

This theorem is used by: (None)