Theorem isusgren 15913

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 15911	. . 3 ⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}}
2	1	eleq2i 2274	. 2 ⊢ (𝐺 ∈ USGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}})
3		fveq2 5600	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuspgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2258	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 4900	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2211	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 4899	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2256	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 5600	. . . . . . 7 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
11		isuspgr.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	eqtr4di 2258	. . . . . 6 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 3632	. . . . 5 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	rabeqdv 2771	. . . 4 ⊢ (ℎ = 𝐺 → {𝑥 ∈ 𝒫 (Vtx‘ℎ) ∣ 𝑥 ≈ 2o} = {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2o})
15	5, 9, 14	f1eq123d 5537	. . 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 15778	. . . . . . 7 ⊢ (𝑔 ∈ V → (Vtx‘𝑔) ∈ V)
17	16	elv 2781	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
18	17	a1i 9	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
19		fveq2 5600	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
20		iedgex 15779	. . . . . . . 8 ⊢ (𝑔 ∈ V → (iEdg‘𝑔) ∈ V)
21	20	elv 2781	. . . . . . 7 ⊢ (iEdg‘𝑔) ∈ V
22	21	a1i 9	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
23		fveq2 5600	. . . . . . 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 4900	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → dom 𝑒 = dom (iEdg‘ℎ))
27		pweq 3630	. . . . . . . . 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 2771	. . . . . . 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 5537	. . . . . 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 3044	. . . . 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 3044	. . . 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 2332	. . 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 2928	. 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 1373   ∈ wcel 2178  {cab 2193  {crab 2490  Vcvv 2777  [wsbc 3006  𝒫 cpw 3627   class class class wbr 4060  dom cdm 4694  –1-1→wf1 5288  ‘cfv 5291  2_oc2o 6521   ≈ cen 6850  Vtxcvtx 15772  iEdgciedg 15773  USGraphcusgr 15909

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-cnre 8073

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-if 3581  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-sub 8282  df-inn 9074  df-2 9132  df-3 9133  df-4 9134  df-5 9135  df-6 9136  df-7 9137  df-8 9138  df-9 9139  df-n0 9333  df-dec 9542  df-ndx 12996  df-slot 12997  df-base 12999  df-edgf 15765  df-vtx 15774  df-iedg 15775  df-usgren 15911

This theorem is referenced by:  usgrfen  15915  isusgropen  15920  ausgrusgrben  15923  ausgrusgrien  15926  usgruspgr  15938  usgrumgruspgr  15940  usgruspgrben  15941  usgrislfuspgrdom  15945