Theorem subgrprop 29305

Description: The properties of a subgraph. (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
subgrprop	⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))

Proof of Theorem subgrprop

Step	Hyp	Ref	Expression
1		subgrv 29302	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
2		issubgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝑆)
3		issubgr.a	. . . . 5 ⊢ 𝐴 = (Vtx‘𝐺)
4		issubgr.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝑆)
5		issubgr.b	. . . . 5 ⊢ 𝐵 = (iEdg‘𝐺)
6		issubgr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝑆)
7	2, 3, 4, 5, 6	issubgr 29303	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
8	7	biimpd 229	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
9	8	ancoms 458	. 2 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
10	1, 9	mpcom 38	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689   ↾ cres 5691  ‘cfv 6563  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079   SubGraph csubgr 29299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701  df-iota 6516  df-fv 6571  df-subgr 29300

This theorem is referenced by:  subgrprop2  29306  subgrwlk  35117