Theorem subgrprop2 27641

Description: The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺, connecting vertices of the subgraph only. (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
issubgr.v	⊢ 𝑉 = (Vtx‘𝑆)
issubgr.a	⊢ 𝐴 = (Vtx‘𝐺)
issubgr.i	⊢ 𝐼 = (iEdg‘𝑆)
issubgr.b	⊢ 𝐵 = (iEdg‘𝐺)
issubgr.e	⊢ 𝐸 = (Edg‘𝑆)

Assertion

Ref	Expression
subgrprop2	⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))

Proof of Theorem subgrprop2

Step	Hyp	Ref	Expression
1		issubgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝑆)
2		issubgr.a	. . 3 ⊢ 𝐴 = (Vtx‘𝐺)
3		issubgr.i	. . 3 ⊢ 𝐼 = (iEdg‘𝑆)
4		issubgr.b	. . 3 ⊢ 𝐵 = (iEdg‘𝐺)
5		issubgr.e	. . 3 ⊢ 𝐸 = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop 27640	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7		resss 5916	. . . 4 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8		sseq1 3946	. . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
9	7, 8	mpbiri 257	. . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵)
10	9	3anim2i 1152	. 2 ⊢ ((𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))
11	6, 10	syl 17	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074  dom cdm 5589   ↾ cres 5591  ‘cfv 6433  Vtxcvtx 27366  iEdgciedg 27367  Edgcedg 27417   SubGraph csubgr 27634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-res 5601  df-iota 6391  df-fv 6441  df-subgr 27635

This theorem is referenced by:  uhgrissubgr  27642  subgrprop3  27643  subgrfun  27648  subgreldmiedg  27650  subgruhgredgd  27651  subumgredg2  27652  subuhgr  27653  subupgr  27654  subumgr  27655  subusgr  27656  subgrwlk  33094