Theorem issubgr2 29307

Description: The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-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
issubgr2	⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)))

Proof of Theorem issubgr2

Step	Hyp	Ref	Expression
1		issubgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝑆)
2		issubgr.a	. . . 4 ⊢ 𝐴 = (Vtx‘𝐺)
3		issubgr.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
4		issubgr.b	. . . 4 ⊢ 𝐵 = (iEdg‘𝐺)
5		issubgr.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
6	1, 2, 3, 4, 5	issubgr 29306	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
7	6	3adant2 1131	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
8		resss 6031	. . . . 5 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵
9		sseq1 4034	. . . . 5 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
10	8, 9	mpbiri 258	. . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵)
11		funssres 6622	. . . . . . 7 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
12	11	eqcomd 2746	. . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼))
13	12	ex 412	. . . . 5 ⊢ (Fun 𝐵 → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼)))
14	13	3ad2ant2 1134	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼)))
15	10, 14	impbid2 226	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼 ⊆ 𝐵))
16	15	3anbi2d 1441	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → ((𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)))
17	7, 16	bitrd 279	1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166  dom cdm 5700   ↾ cres 5702  Fun wfun 6567  ‘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-12 2178  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-mo 2543  df-eu 2572  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-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-subgr 29303

This theorem is referenced by:  uhgrspansubgr  29326