Theorem isusgr 29089

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 29087	. . 3 ⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
2	1	eleq2i 2818	. 2 ⊢ (𝐺 ∈ USGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}})
3		fveq2 6901	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuspgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2784	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 5912	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2735	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 5911	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2782	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 6901	. . . . . . . 8 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
11		isuspgr.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	eqtr4di 2784	. . . . . . 7 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 4624	. . . . . 6 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	difeq1d 4120	. . . . 5 ⊢ (ℎ = 𝐺 → (𝒫 (Vtx‘ℎ) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
15	14	rabeqdv 3435	. . . 4 ⊢ (ℎ = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
16	5, 9, 15	f1eq123d 6835	. . 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 6916	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
18		fveq2 6901	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
19		fvexd 6916	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
20		fveq2 6901	. . . . . . 7 ⊢ (𝑔 = ℎ → (iEdg‘𝑔) = (iEdg‘ℎ))
21	20	adantr 479	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) = (iEdg‘ℎ))
22		simpr 483	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝑒 = (iEdg‘ℎ))
23	22	dmeqd 5912	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → dom 𝑒 = dom (iEdg‘ℎ))
24		pweq 4621	. . . . . . . . . 10 ⊢ (𝑣 = (Vtx‘ℎ) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
25	24	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
26	25	difeq1d 4120	. . . . . . . 8 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘ℎ) ∖ {∅}))
27	26	rabeqdv 3435	. . . . . . 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 6835	. . . . . 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 3824	. . . . 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 3824	. . . 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 2799	. . 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 3668	. 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 282	1 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  {cab 2703  {crab 3419  Vcvv 3462  [wsbc 3776   ∖ cdif 3944  ∅c0 4325  𝒫 cpw 4607  {csn 4633  dom cdm 5682  –1-1→wf1 6551  ‘cfv 6554  2c2 12319  ♯chash 14347  Vtxcvtx 28932  iEdgciedg 28933  USGraphcusgr 29085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fv 6562  df-usgr 29087

This theorem is referenced by:  usgrf  29091  isusgrs  29092  usgruspgr  29116  usgrumgruspgr  29118  usgrislfuspgr  29123  usgr0e  29172  usgr0  29179

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