Theorem issubgr 16181

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 5648	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Vtx‘𝑠) = (Vtx‘𝑆))
2	1	adantr 276	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑠) = (Vtx‘𝑆))
3		fveq2 5648	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	adantl 277	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = (Vtx‘𝐺))
5	2, 4	sseq12d 3259	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)))
6		fveq2 5648	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (iEdg‘𝑠) = (iEdg‘𝑆))
7	6	adantr 276	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑠) = (iEdg‘𝑆))
8		fveq2 5648	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
9	8	adantl 277	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑔 = 𝐺) → (iEdg‘𝑔) = (iEdg‘𝐺))
10	6	dmeqd 4939	. . . . . . . 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 5020	. . . . . 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 5648	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Edg‘𝑠) = (Edg‘𝑆))
15	1	pweqd 3661	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 (Vtx‘𝑠) = 𝒫 (Vtx‘𝑆))
16	14, 15	sseq12d 3259	. . . . . 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 1349	. . . 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 16178	. . . 4 ⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
20	18, 19	brabga 4364	. . 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 3256	. . 3 ⊢ (𝑉 ⊆ 𝐴 ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
25		issubgr.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
26		issubgr.b	. . . . 5 ⊢ 𝐵 = (iEdg‘𝐺)
27	25	dmeqi 4938	. . . . 5 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
28	26, 27	reseq12i 5017	. . . 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 3660	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
32	30, 31	sseq12i 3256	. . 3 ⊢ (𝐸 ⊆ 𝒫 𝑉 ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
33	24, 29, 32	3anbi123i 1215	. 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 1005   = wceq 1398   ∈ wcel 2202   ⊆ wss 3201  𝒫 cpw 3656   class class class wbr 4093  dom cdm 4731   ↾ cres 4733  ‘cfv 5333  Vtxcvtx 15936  iEdgciedg 15937  Edgcedg 15981   SubGraph csubgr 16177

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-dm 4741  df-res 4743  df-iota 5293  df-fv 5341  df-subgr 16178

This theorem is referenced by:  issubgr2  16182  subgrprop  16183  uhgrissubgr  16185  egrsubgr  16187  0grsubgr  16188