Theorem usgrstrrepeen 15994

Description: Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.)

Hypotheses

Ref	Expression
usgrstrrepe.v	⊢ 𝑉 = (Base‘𝐺)
usgrstrrepe.i	⊢ 𝐼 = (.ef‘ndx)
usgrstrrepe.s	⊢ (𝜑 → 𝐺 Struct 𝑋)
usgrstrrepe.b	⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)
usgrstrrepe.w	⊢ (𝜑 → 𝐸 ∈ 𝑊)
usgrstrrepeen.e	⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o})

Assertion

Ref	Expression
usgrstrrepeen	⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph)

Distinct variable groups:   𝑥,𝐺   𝑥,𝐸   𝑥,𝐼   𝑥,𝑉   𝜑,𝑥

Allowed substitution hints:   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem usgrstrrepeen

Step	Hyp	Ref	Expression
1		usgrstrrepeen.e	. . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o})
2		usgrstrrepe.i	. . . . . . . . 9 ⊢ 𝐼 = (.ef‘ndx)
3		usgrstrrepe.s	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Struct 𝑋)
4		usgrstrrepe.b	. . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)
5		usgrstrrepe.w	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
6	2, 3, 4, 5	setsvtx 15817	. . . . . . . 8 ⊢ (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Base‘𝐺))
7		usgrstrrepe.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝐺)
8	6, 7	eqtr4di 2260	. . . . . . 7 ⊢ (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝑉)
9	8	pweqd 3634	. . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝒫 𝑉)
10	9	rabeqdv 2773	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o} = {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o})
11		f1eq3 5504	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o} = {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o} → (𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o}))
12	10, 11	syl 14	. . . 4 ⊢ (𝜑 → (𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o}))
13	1, 12	mpbird 167	. . 3 ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o})
14	2, 3, 4, 5	setsiedg 15818	. . . 4 ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸)
15	14	dmeqd 4902	. . . 4 ⊢ (𝜑 → dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = dom 𝐸)
16		eqidd 2210	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o} = {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o})
17	14, 15, 16	f1eq123d 5540	. . 3 ⊢ (𝜑 → ((iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)):dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o}))
18	13, 17	mpbird 167	. 2 ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)):dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o})
19		structex 13010	. . . . 5 ⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V)
20	3, 19	syl 14	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
21		edgfndxnn 15774	. . . . . 6 ⊢ (.ef‘ndx) ∈ ℕ
22	2, 21	eqeltri 2282	. . . . 5 ⊢ 𝐼 ∈ ℕ
23	22	a1i 9	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ℕ)
24		setsex 13030	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐼 ∈ ℕ ∧ 𝐸 ∈ 𝑊) → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V)
25	20, 23, 5, 24	syl3anc 1252	. . 3 ⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V)
26		eqid 2209	. . . 4 ⊢ (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩))
27		eqid 2209	. . . 4 ⊢ (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))
28	26, 27	isusgren 15921	. . 3 ⊢ ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V → ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph ↔ (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)):dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o}))
29	25, 28	syl 14	. 2 ⊢ (𝜑 → ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph ↔ (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)):dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∣ 𝑥 ≈ 2o}))
30	18, 29	mpbird 167	1 ⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1375   ∈ wcel 2180  {crab 2492  Vcvv 2779  𝒫 cpw 3629  ⟨cop 3649   class class class wbr 4062  dom cdm 4696  –1-1→wf1 5291  ‘cfv 5294  (class class class)co 5974  2_oc2o 6526   ≈ cen 6855  ℕcn 9078   Struct cstr 12994  ndxcnx 12995   sSet csts 12996  Basecbs 12998  .efcedgf 15770  Vtxcvtx 15778  iEdgciedg 15779  USGraphcusgr 15917

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-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-ltadd 8083  ax-pre-mulgt0 8084

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-1o 6532  df-2o 6533  df-en 6858  df-dom 6859  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-z 9415  df-dec 9547  df-struct 13000  df-ndx 13001  df-slot 13002  df-base 13004  df-sets 13005  df-edgf 15771  df-vtx 15780  df-iedg 15781  df-usgren 15919

This theorem is referenced by: (None)