Theorem isuhgr 27411

Description: The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
isuhgr	⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Proof of Theorem isuhgr

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

Step	Hyp	Ref	Expression
1		df-uhgr 27409	. . 3 ⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
2	1	eleq2i 2831	. 2 ⊢ (𝐺 ∈ UHGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})})
3		fveq2 6768	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuhgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2797	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 5811	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2748	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 5810	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2795	. . . 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 2797	. . . . . 6 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 4557	. . . . 5 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	difeq1d 4060	. . . 4 ⊢ (ℎ = 𝐺 → (𝒫 (Vtx‘ℎ) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
15	5, 9, 14	feq123d 6585	. . 3 ⊢ (ℎ = 𝐺 → ((iEdg‘ℎ):dom (iEdg‘ℎ)⟶(𝒫 (Vtx‘ℎ) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(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 480	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) = (iEdg‘ℎ))
21		simpr 484	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝑒 = (iEdg‘ℎ))
22	21	dmeqd 5811	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → dom 𝑒 = dom (iEdg‘ℎ))
23		simpr 484	. . . . . . . . . 10 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → 𝑣 = (Vtx‘ℎ))
24	23	pweqd 4557	. . . . . . . . 9 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
25	24	difeq1d 4060	. . . . . . . 8 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘ℎ) ∖ {∅}))
26	25	adantr 480	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘ℎ) ∖ {∅}))
27	21, 22, 26	feq123d 6585	. . . . . 6 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)⟶(𝒫 (Vtx‘ℎ) ∖ {∅})))
28	18, 20, 27	sbcied2 3766	. . . . 5 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)⟶(𝒫 (Vtx‘ℎ) ∖ {∅})))
29	16, 17, 28	sbcied2 3766	. . . 4 ⊢ (𝑔 = ℎ → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)⟶(𝒫 (Vtx‘ℎ) ∖ {∅})))
30	29	cbvabv 2812	. . 3 ⊢ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} = {ℎ ∣ (iEdg‘ℎ):dom (iEdg‘ℎ)⟶(𝒫 (Vtx‘ℎ) ∖ {∅})}
31	15, 30	elab2g 3612	. 2 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
32	2, 31	syl5bb 282	1 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  {cab 2716  Vcvv 3430  [wsbc 3719   ∖ cdif 3888  ∅c0 4261  𝒫 cpw 4538  {csn 4566  dom cdm 5588  ⟶wf 6426  ‘cfv 6430  Vtxcvtx 27347  iEdgciedg 27348  UHGraphcuhgr 27407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-uhgr 27409

This theorem is referenced by:  uhgrf  27413  uhgreq12g  27416  ushgruhgr  27420  isuhgrop  27421  uhgr0e  27422  uhgr0  27424  uhgrun  27425  uhgrstrrepe  27429  incistruhgr  27430  upgruhgr  27453  subuhgr  27634