Theorem isubgrupgr 48771

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 29543	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2		isubgrupgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	isubgrsubgr 48770	. . 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 29731	. 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 1570   ∈ wcel 2145   ⊆ wss 3902   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Vtxcvtx 29437  UHGraphcuhgr 29497  UPGraphcupgr 29521   SubGraph csubgr 29711   ISubGr cisubgr 48761

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-vtx 29439  df-iedg 29440  df-edg 29489  df-uhgr 29499  df-upgr 29523  df-subgr 29712  df-isubgr 48762

This theorem is used by: (None)