Theorem isuspgr 28409

Description: The property of being a simple pseudograph. (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
isuspgr	⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USPGraph ↔ 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 isuspgr

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

Step	Hyp	Ref	Expression
1		df-uspgr 28407	. . 3 ⊢ USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
2	1	eleq2i 2825	. 2 ⊢ (𝐺 ∈ USPGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}})
3		fveq2 6891	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuspgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2790	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 5905	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2741	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 5904	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2788	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 6891	. . . . . . . 8 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
11		isuspgr.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	eqtr4di 2790	. . . . . . 7 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 4619	. . . . . 6 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	difeq1d 4121	. . . . 5 ⊢ (ℎ = 𝐺 → (𝒫 (Vtx‘ℎ) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
15	14	rabeqdv 3447	. . . 4 ⊢ (ℎ = 𝐺 → {𝑥 ∈ (𝒫 (Vtx‘ℎ) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
16	5, 9, 15	f1eq123d 6825	. . 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 6906	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
18		fveq2 6891	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
19		fvexd 6906	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
20		fveq2 6891	. . . . . . 7 ⊢ (𝑔 = ℎ → (iEdg‘𝑔) = (iEdg‘ℎ))
21	20	adantr 481	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) = (iEdg‘ℎ))
22		simpr 485	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝑒 = (iEdg‘ℎ))
23	22	dmeqd 5905	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → dom 𝑒 = dom (iEdg‘ℎ))
24		pweq 4616	. . . . . . . . . 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 4121	. . . . . . . 8 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘ℎ) ∖ {∅}))
27	26	rabeqdv 3447	. . . . . . 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 6825	. . . . . 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 2805	. . 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 3670	. 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 ∈ USPGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  {crab 3432  Vcvv 3474  [wsbc 3777   ∖ cdif 3945  ∅c0 4322  𝒫 cpw 4602  {csn 4628   class class class wbr 5148  dom cdm 5676  –1-1→wf1 6540  ‘cfv 6543   ≤ cle 11248  2c2 12266  ♯chash 14289  Vtxcvtx 28253  iEdgciedg 28254  USPGraphcuspgr 28405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551  df-uspgr 28407

This theorem is referenced by:  uspgrf  28411  isuspgrop  28418  uspgrushgr  28432  uspgrupgr  28433  uspgrupgrushgr  28434  usgruspgr  28435  uspgr1e  28498