Theorem isusgren 16002

Description: The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)

Hypotheses

Ref	Expression
isuspgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isuspgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
isusgren	⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o}))

Distinct variable groups:   𝑥,𝐺   𝑥,𝑉

Allowed substitution hints:   𝑈(𝑥)   𝐸(𝑥)

Proof of Theorem isusgren

Dummy variables 𝑒 𝑔 ℎ 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-usgren 16000	. . 3 ⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}}
2	1	eleq2i 2296	. 2 ⊢ (𝐺 ∈ USGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}})
3		fveq2 5635	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuspgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2280	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 4931	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2233	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 4930	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2278	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 5635	. . . . . . 7 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
11		isuspgr.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	eqtr4di 2280	. . . . . 6 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 3655	. . . . 5 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	rabeqdv 2794	. . . 4 ⊢ (ℎ = 𝐺 → {𝑥 ∈ 𝒫 (Vtx‘ℎ) ∣ 𝑥 ≈ 2o} = {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o})
15	5, 9, 14	f1eq123d 5572	. . 3 ⊢ (ℎ = 𝐺 → ((iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ 𝒫 (Vtx‘ℎ) ∣ 𝑥 ≈ 2o} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o}))
16		vtxex 15862	. . . . . . 7 ⊢ (𝑔 ∈ V → (Vtx‘𝑔) ∈ V)
17	16	elv 2804	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
18	17	a1i 9	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
19		fveq2 5635	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
20		iedgex 15863	. . . . . . . 8 ⊢ (𝑔 ∈ V → (iEdg‘𝑔) ∈ V)
21	20	elv 2804	. . . . . . 7 ⊢ (iEdg‘𝑔) ∈ V
22	21	a1i 9	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
23		fveq2 5635	. . . . . . 7 ⊢ (𝑔 = ℎ → (iEdg‘𝑔) = (iEdg‘ℎ))
24	23	adantr 276	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) = (iEdg‘ℎ))
25		simpr 110	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝑒 = (iEdg‘ℎ))
26	25	dmeqd 4931	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → dom 𝑒 = dom (iEdg‘ℎ))
27		pweq 3653	. . . . . . . . 9 ⊢ (𝑣 = (Vtx‘ℎ) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
28	27	ad2antlr 489	. . . . . . . 8 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
29	28	rabeqdv 2794	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → {𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o} = {𝑥 ∈ 𝒫 (Vtx‘ℎ) ∣ 𝑥 ≈ 2o})
30	25, 26, 29	f1eq123d 5572	. . . . . 6 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ 𝒫 (Vtx‘ℎ) ∣ 𝑥 ≈ 2o}))
31	22, 24, 30	sbcied2 3067	. . . . 5 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ 𝒫 (Vtx‘ℎ) ∣ 𝑥 ≈ 2o}))
32	18, 19, 31	sbcied2 3067	. . . 4 ⊢ (𝑔 = ℎ → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ 𝒫 (Vtx‘ℎ) ∣ 𝑥 ≈ 2o}))
33	32	cbvabv 2354	. . 3 ⊢ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}} = {ℎ ∣ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ 𝒫 (Vtx‘ℎ) ∣ 𝑥 ≈ 2o}}
34	15, 33	elab2g 2951	. 2 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o}))
35	2, 34	bitrid 192	1 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215  {crab 2512  Vcvv 2800  [wsbc 3029  𝒫 cpw 3650   class class class wbr 4086  dom cdm 4723  –1-1→wf1 5321  ‘cfv 5324  2_oc2o 6571   ≈ cen 6902  Vtxcvtx 15856  iEdgciedg 15857  USGraphcusgr 15998

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-sub 8345  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-dec 9605  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-usgren 16000

This theorem is referenced by:  usgrfen  16004  isusgropen  16009  ausgrusgrben  16012  ausgrusgrien  16015  usgruspgr  16027  usgrumgruspgr  16029  usgruspgrben  16030  usgrislfuspgrdom  16034  usgrstrrepeen  16075  usgr0e  16076  usgr0  16083