Theorem subgrprop2 29343

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 29342	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7		resss 5966	. . . 4 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8		sseq1 3947	. . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
9	7, 8	mpbiri 258	. . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵)
10	9	3anim2i 1154	. 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 1087   = wceq 1542   ⊆ wss 3889  𝒫 cpw 4541   class class class wbr 5085  dom cdm 5631   ↾ cres 5633  ‘cfv 6498  Vtxcvtx 29065  iEdgciedg 29066  Edgcedg 29116   SubGraph csubgr 29336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dm 5641  df-res 5643  df-iota 6454  df-fv 6506  df-subgr 29337

This theorem is referenced by:  uhgrissubgr  29344  subgrprop3  29345  subgrfun  29350  subgreldmiedg  29352  subgruhgredgd  29353  subumgredg2  29354  subuhgr  29355  subupgr  29356  subumgr  29357  subusgr  29358  subgrwlk  35314