Theorem issubgr 16107

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 5639	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Vtx‘𝑠) = (Vtx‘𝑆))
2	1	adantr 276	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑠) = (Vtx‘𝑆))
3		fveq2 5639	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	adantl 277	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = (Vtx‘𝐺))
5	2, 4	sseq12d 3258	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)))
6		fveq2 5639	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (iEdg‘𝑠) = (iEdg‘𝑆))
7	6	adantr 276	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑠) = (iEdg‘𝑆))
8		fveq2 5639	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
9	8	adantl 277	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑔) = (iEdg‘𝐺))
10	6	dmeqd 4933	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
11	10	adantr 276	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
12	9, 11	reseq12d 5014	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
13	7, 12	eqeq12d 2246	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
14		fveq2 5639	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Edg‘𝑠) = (Edg‘𝑆))
15	1	pweqd 3657	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 (Vtx‘𝑠) = 𝒫 (Vtx‘𝑆))
16	14, 15	sseq12d 3258	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
17	16	adantr 276	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
18	5, 13, 17	3anbi123d 1348	. . . 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 16104	. . . 4 ⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
20	18, 19	brabga 4358	. . 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 268	. 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 3255	. . 3 ⊢ (𝑉 ⊆ 𝐴 ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
25		issubgr.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
26		issubgr.b	. . . . 5 ⊢ 𝐵 = (iEdg‘𝐺)
27	25	dmeqi 4932	. . . . 5 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
28	26, 27	reseq12i 5011	. . . 4 ⊢ (𝐵 ↾ dom 𝐼) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))
29	25, 28	eqeq12i 2245	. . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
30		issubgr.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
31	22	pweqi 3656	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
32	30, 31	sseq12i 3255	. . 3 ⊢ (𝐸 ⊆ 𝒫 𝑉 ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
33	24, 29, 32	3anbi123i 1214	. 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 198	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202   ⊆ wss 3200  𝒫 cpw 3652   class class class wbr 4088  dom cdm 4725   ↾ cres 4727  ‘cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907   SubGraph csubgr 16103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-dm 4735  df-res 4737  df-iota 5286  df-fv 5334  df-subgr 16104

This theorem is referenced by:  issubgr2  16108  subgrprop  16109  uhgrissubgr  16111  egrsubgr  16113  0grsubgr  16114