Theorem issubgr 29306

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 6920	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Vtx‘𝑠) = (Vtx‘𝑆))
2	1	adantr 480	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑠) = (Vtx‘𝑆))
3		fveq2 6920	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	adantl 481	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = (Vtx‘𝐺))
5	2, 4	sseq12d 4042	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)))
6		fveq2 6920	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (iEdg‘𝑠) = (iEdg‘𝑆))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑠) = (iEdg‘𝑆))
8		fveq2 6920	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
9	8	adantl 481	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑔) = (iEdg‘𝐺))
10	6	dmeqd 5930	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
11	10	adantr 480	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
12	9, 11	reseq12d 6010	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
13	7, 12	eqeq12d 2756	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
14		fveq2 6920	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Edg‘𝑠) = (Edg‘𝑆))
15	1	pweqd 4639	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 (Vtx‘𝑠) = 𝒫 (Vtx‘𝑆))
16	14, 15	sseq12d 4042	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
17	16	adantr 480	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
18	5, 13, 17	3anbi123d 1436	. . . 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 29303	. . . 4 ⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
20	18, 19	brabga 5553	. . 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 458	. 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 4039	. . 3 ⊢ (𝑉 ⊆ 𝐴 ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
25		issubgr.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
26		issubgr.b	. . . . 5 ⊢ 𝐵 = (iEdg‘𝐺)
27	25	dmeqi 5929	. . . . 5 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
28	26, 27	reseq12i 6007	. . . 4 ⊢ (𝐵 ↾ dom 𝐼) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))
29	25, 28	eqeq12i 2758	. . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
30		issubgr.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
31	22	pweqi 4638	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
32	30, 31	sseq12i 4039	. . 3 ⊢ (𝐸 ⊆ 𝒫 𝑉 ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
33	24, 29, 32	3anbi123i 1155	. 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 289	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 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  ‘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-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-dm 5710  df-res 5712  df-iota 6525  df-fv 6581  df-subgr 29303

This theorem is referenced by:  issubgr2  29307  subgrprop  29308  uhgrissubgr  29310  egrsubgr  29312  0grsubgr  29313  uhgrspan1  29338