Theorem issubgr2 29682

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 29681	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
7	6	3adant2 1149	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
8		resss 6002	. . . . 5 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵
9		sseq1 3963	. . . . 5 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
10	8, 9	mpbiri 261	. . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵)
11		funssres 6584	. . . . . . 7 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
12	11	eqcomd 2771	. . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼))
13	12	ex 418	. . . . 5 ⊢ (Fun 𝐵 → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼)))
14	13	3ad2ant2 1152	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼)))
15	10, 14	impbid2 229	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼 ⊆ 𝐵))
16	15	3anbi2d 1469	. 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 282	1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  𝒫 cpw 4564   class class class wbr 5111  dom cdm 5663   ↾ cres 5665  Fun wfun 6534  ‘cfv 6540  Vtxcvtx 29403  iEdgciedg 29404  Edgcedg 29454   SubGraph csubgr 29677

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-subgr 29678

This theorem is used by:  uhgrspansubgr  29701