Theorem issubgr 29198

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 6858	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Vtx‘𝑠) = (Vtx‘𝑆))
2	1	adantr 480	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑠) = (Vtx‘𝑆))
3		fveq2 6858	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	adantl 481	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = (Vtx‘𝐺))
5	2, 4	sseq12d 3980	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)))
6		fveq2 6858	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (iEdg‘𝑠) = (iEdg‘𝑆))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑠) = (iEdg‘𝑆))
8		fveq2 6858	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
9	8	adantl 481	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑔) = (iEdg‘𝐺))
10	6	dmeqd 5869	. . . . . . . 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 5951	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
13	7, 12	eqeq12d 2745	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
14		fveq2 6858	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Edg‘𝑠) = (Edg‘𝑆))
15	1	pweqd 4580	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 (Vtx‘𝑠) = 𝒫 (Vtx‘𝑆))
16	14, 15	sseq12d 3980	. . . . . 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 1438	. . . 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 29195	. . . 4 ⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
20	18, 19	brabga 5494	. . 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 3977	. . 3 ⊢ (𝑉 ⊆ 𝐴 ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
25		issubgr.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
26		issubgr.b	. . . . 5 ⊢ 𝐵 = (iEdg‘𝐺)
27	25	dmeqi 5868	. . . . 5 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
28	26, 27	reseq12i 5948	. . . 4 ⊢ (𝐵 ↾ dom 𝐼) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))
29	25, 28	eqeq12i 2747	. . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
30		issubgr.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
31	22	pweqi 4579	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
32	30, 31	sseq12i 3977	. . 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 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  𝒫 cpw 4563   class class class wbr 5107  dom cdm 5638   ↾ cres 5640  ‘cfv 6511  Vtxcvtx 28923  iEdgciedg 28924  Edgcedg 28974   SubGraph csubgr 29194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-dm 5648  df-res 5650  df-iota 6464  df-fv 6519  df-subgr 29195

This theorem is referenced by:  issubgr2  29199  subgrprop  29200  uhgrissubgr  29202  egrsubgr  29204  0grsubgr  29205  uhgrspan1  29230