Theorem isushgr 27420

Description: The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
isushgr	⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})))

Proof of Theorem isushgr

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

Step	Hyp	Ref	Expression
1		df-ushgr 27418	. . 3 ⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
2	1	eleq2i 2830	. 2 ⊢ (𝐺 ∈ USHGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})})
3		fveq2 6768	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuhgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2796	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 5809	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2747	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 5808	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2794	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 6768	. . . . . . 7 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
11		isuhgr.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	eqtr4di 2796	. . . . . 6 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 4554	. . . . 5 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	difeq1d 4057	. . . 4 ⊢ (ℎ = 𝐺 → (𝒫 (Vtx‘ℎ) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
15	5, 9, 14	f1eq123d 6702	. . 3 ⊢ (ℎ = 𝐺 → ((iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→(𝒫 (Vtx‘ℎ) ∖ {∅}) ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})))
16		fvexd 6783	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
17		fveq2 6768	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
18		fvexd 6783	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
19		fveq2 6768	. . . . . . 7 ⊢ (𝑔 = ℎ → (iEdg‘𝑔) = (iEdg‘ℎ))
20	19	adantr 481	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) = (iEdg‘ℎ))
21		simpr 485	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝑒 = (iEdg‘ℎ))
22	21	dmeqd 5809	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → dom 𝑒 = dom (iEdg‘ℎ))
23		simpr 485	. . . . . . . . . 10 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → 𝑣 = (Vtx‘ℎ))
24	23	pweqd 4554	. . . . . . . . 9 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
25	24	difeq1d 4057	. . . . . . . 8 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘ℎ) ∖ {∅}))
26	25	adantr 481	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘ℎ) ∖ {∅}))
27	21, 22, 26	f1eq123d 6702	. . . . . 6 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→(𝒫 (Vtx‘ℎ) ∖ {∅})))
28	18, 20, 27	sbcied2 3764	. . . . 5 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→(𝒫 (Vtx‘ℎ) ∖ {∅})))
29	16, 17, 28	sbcied2 3764	. . . 4 ⊢ (𝑔 = ℎ → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→(𝒫 (Vtx‘ℎ) ∖ {∅})))
30	29	cbvabv 2811	. . 3 ⊢ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} = {ℎ ∣ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→(𝒫 (Vtx‘ℎ) ∖ {∅})}
31	15, 30	elab2g 3612	. 2 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})))
32	2, 31	syl5bb 283	1 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  Vcvv 3431  [wsbc 3717   ∖ cdif 3885  ∅c0 4258  𝒫 cpw 4535  {csn 4563  dom cdm 5586  –1-1→wf1 6425  ‘cfv 6428  Vtxcvtx 27355  iEdgciedg 27356  USHGraphcushgr 27416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5076  df-opab 5138  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fv 6436  df-ushgr 27418

This theorem is referenced by:  ushgrf  27422  uspgrushgr  27534