Theorem subgrprop 29308

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 29305	. 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 29306	. . . 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 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166  dom cdm 5700   ↾ cres 5702  ‘cfv 6573  Vtxcvtx 29031  iEdgciedg 29032  Edgcedg 29082   SubGraph csubgr 29302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-res 5712  df-iota 6525  df-fv 6581  df-subgr 29303

This theorem is referenced by:  subgrprop2  29309  subgrwlk  35100