Theorem isusgr 29170

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
isusgr	⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))

Distinct variable groups:   𝑥,𝐺   𝑥,𝑉

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

Proof of Theorem isusgr

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

Step	Hyp	Ref	Expression
1		df-usgr 29168	. . 3 ⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
2	1	eleq2i 2833	. 2 ⊢ (𝐺 ∈ USGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}})
3		fveq2 6906	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuspgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2795	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 5916	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2746	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 5915	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2793	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 6906	. . . . . . . 8 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
11		isuspgr.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	eqtr4di 2795	. . . . . . 7 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 4617	. . . . . 6 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	difeq1d 4125	. . . . 5 ⊢ (ℎ = 𝐺 → (𝒫 (Vtx‘ℎ) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
15	14	rabeqdv 3452	. . . 4 ⊢ (ℎ = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
16	5, 9, 15	f1eq123d 6840	. . 3 ⊢ (ℎ = 𝐺 → ((iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
17		fvexd 6921	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
18		fveq2 6906	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
19		fvexd 6921	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
20		fveq2 6906	. . . . . . 7 ⊢ (𝑔 = ℎ → (iEdg‘𝑔) = (iEdg‘ℎ))
21	20	adantr 480	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) = (iEdg‘ℎ))
22		simpr 484	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝑒 = (iEdg‘ℎ))
23	22	dmeqd 5916	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → dom 𝑒 = dom (iEdg‘ℎ))
24		pweq 4614	. . . . . . . . . 10 ⊢ (𝑣 = (Vtx‘ℎ) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
25	24	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
26	25	difeq1d 4125	. . . . . . . 8 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘ℎ) ∖ {∅}))
27	26	rabeqdv 3452	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) = 2})
28	22, 23, 27	f1eq123d 6840	. . . . . 6 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
29	19, 21, 28	sbcied2 3833	. . . . 5 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
30	17, 18, 29	sbcied2 3833	. . . 4 ⊢ (𝑔 = ℎ → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
31	30	cbvabv 2812	. . 3 ⊢ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} = {ℎ ∣ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) = 2}}
32	16, 31	elab2g 3680	. 2 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
33	2, 32	bitrid 283	1 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  {crab 3436  Vcvv 3480  [wsbc 3788   ∖ cdif 3948  ∅c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  –1-1→wf1 6558  ‘cfv 6561  2c2 12321  ♯chash 14369  Vtxcvtx 29013  iEdgciedg 29014  USGraphcusgr 29166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-usgr 29168

This theorem is referenced by:  usgrf  29172  isusgrs  29173  usgruspgr  29197  usgrumgruspgr  29199  usgrislfuspgr  29204  usgr0e  29253  usgr0  29260