Theorem isuhgr 29049

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 29047	. . 3 ⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
2	1	eleq2i 2825	. 2 ⊢ (𝐺 ∈ UHGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})})
3		fveq2 6831	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuhgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2786	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 5852	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2742	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 5851	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2784	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 6831	. . . . . . 7 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
11		isuhgr.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	eqtr4di 2786	. . . . . 6 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 4568	. . . . 5 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	difeq1d 4076	. . . 4 ⊢ (ℎ = 𝐺 → (𝒫 (Vtx‘ℎ) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
15	5, 9, 14	feq123d 6648	. . 3 ⊢ (ℎ = 𝐺 → ((iEdg‘ℎ):dom (iEdg‘ℎ)⟶(𝒫 (Vtx‘ℎ) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
16		fvexd 6846	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
17		fveq2 6831	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
18		fvexd 6846	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
19		fveq2 6831	. . . . . . 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 5852	. . . . . . 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 4568	. . . . . . . . 9 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
25	24	difeq1d 4076	. . . . . . . 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 6648	. . . . . 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 3783	. . . . 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 3783	. . . 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 2803	. . 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 3633	. 2 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
32	2, 31	bitrid 283	1 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  Vcvv 3438  [wsbc 3738   ∖ cdif 3896  ∅c0 4284  𝒫 cpw 4551  {csn 4577  dom cdm 5621  ⟶wf 6485  ‘cfv 6489  Vtxcvtx 28985  iEdgciedg 28986  UHGraphcuhgr 29045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-uhgr 29047

This theorem is referenced by:  uhgrf  29051  uhgreq12g  29054  ushgruhgr  29058  isuhgrop  29059  uhgr0e  29060  uhgr0  29062  uhgrun  29063  uhgrstrrepe  29067  incistruhgr  29068  upgruhgr  29091  subuhgr  29275  isubgruhgr  47982  grimuhgr  48001