Theorem issubgr 27205

Description: The property of a set to be a subgraph of another set. (Contributed by AV, 16-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
issubgr	⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))

Proof of Theorem issubgr

Dummy variables 𝑠 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6668	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Vtx‘𝑠) = (Vtx‘𝑆))
2	1	adantr 484	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑠) = (Vtx‘𝑆))
3		fveq2 6668	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	adantl 485	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = (Vtx‘𝐺))
5	2, 4	sseq12d 3908	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)))
6		fveq2 6668	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (iEdg‘𝑠) = (iEdg‘𝑆))
7	6	adantr 484	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑠) = (iEdg‘𝑆))
8		fveq2 6668	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
9	8	adantl 485	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑔) = (iEdg‘𝐺))
10	6	dmeqd 5742	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
11	10	adantr 484	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
12	9, 11	reseq12d 5820	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
13	7, 12	eqeq12d 2754	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
14		fveq2 6668	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Edg‘𝑠) = (Edg‘𝑆))
15	1	pweqd 4504	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 (Vtx‘𝑠) = 𝒫 (Vtx‘𝑆))
16	14, 15	sseq12d 3908	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
17	16	adantr 484	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
18	5, 13, 17	3anbi123d 1437	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠)) ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
19		df-subgr 27202	. . . 4 ⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
20	18, 19	brabga 5386	. . 3 ⊢ ((𝑆 ∈ 𝑈 ∧ 𝐺 ∈ 𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
21	20	ancoms 462	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
22		issubgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝑆)
23		issubgr.a	. . . 4 ⊢ 𝐴 = (Vtx‘𝐺)
24	22, 23	sseq12i 3905	. . 3 ⊢ (𝑉 ⊆ 𝐴 ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
25		issubgr.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
26		issubgr.b	. . . . 5 ⊢ 𝐵 = (iEdg‘𝐺)
27	25	dmeqi 5741	. . . . 5 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
28	26, 27	reseq12i 5817	. . . 4 ⊢ (𝐵 ↾ dom 𝐼) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))
29	25, 28	eqeq12i 2753	. . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
30		issubgr.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
31	22	pweqi 4503	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
32	30, 31	sseq12i 3905	. . 3 ⊢ (𝐸 ⊆ 𝒫 𝑉 ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
33	24, 29, 32	3anbi123i 1156	. 2 ⊢ ((𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
34	21, 33	bitr4di 292	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2113   ⊆ wss 3841  𝒫 cpw 4485   class class class wbr 5027  dom cdm 5519   ↾ cres 5521  ‘cfv 6333  Vtxcvtx 26933  iEdgciedg 26934  Edgcedg 26984   SubGraph csubgr 27201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-xp 5525  df-dm 5529  df-res 5531  df-iota 6291  df-fv 6341  df-subgr 27202

This theorem is referenced by:  issubgr2  27206  subgrprop  27207  uhgrissubgr  27209  egrsubgr  27211  0grsubgr  27212  uhgrspan1  27237