Theorem subgrprop2 29533

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 29532	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7		resss 5991	. . . 4 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8		sseq1 3964	. . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
9	7, 8	mpbiri 261	. . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵)
10	9	3anim2i 1169	. 2 ⊢ ((𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))
11	6, 10	syl 18	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1563   ⊆ wss 3907  𝒫 cpw 4558   class class class wbr 5105  dom cdm 5652   ↾ cres 5654  ‘cfv 6525  Vtxcvtx 29255  iEdgciedg 29256  Edgcedg 29306   SubGraph csubgr 29526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-dm 5662  df-res 5664  df-iota 6481  df-fv 6533  df-subgr 29527

This theorem is referenced by:  uhgrissubgr  29534  subgrprop3  29535  subgrfun  29540  subgreldmiedg  29542  subgruhgredgd  29543  subumgredg2  29544  subuhgr  29545  subupgr  29546  subumgr  29547  subusgr  29548  subgrwlk  35495