Theorem isushgr 29041

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 29039	. . 3 ⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
2	1	eleq2i 2825	. 2 ⊢ (𝐺 ∈ USHGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})})
3		fveq2 6828	. . . . 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 5849	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2742	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 5848	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2784	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 6828	. . . . . . 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 4566	. . . . 5 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	difeq1d 4074	. . . 4 ⊢ (ℎ = 𝐺 → (𝒫 (Vtx‘ℎ) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
15	5, 9, 14	f1eq123d 6760	. . 3 ⊢ (ℎ = 𝐺 → ((iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→(𝒫 (Vtx‘ℎ) ∖ {∅}) ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})))
16		fvexd 6843	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
17		fveq2 6828	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
18		fvexd 6843	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
19		fveq2 6828	. . . . . . 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 5849	. . . . . . 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 4566	. . . . . . . . 9 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
25	24	difeq1d 4074	. . . . . . . 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	f1eq123d 6760	. . . . . 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 3782	. . . . 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 3782	. . . 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 2803	. . 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 3632	. 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	bitrid 283	1 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  Vcvv 3437  [wsbc 3737   ∖ cdif 3895  ∅c0 4282  𝒫 cpw 4549  {csn 4575  dom cdm 5619  –1-1→wf1 6483  ‘cfv 6486  Vtxcvtx 28976  iEdgciedg 28977  USHGraphcushgr 29037

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 5246

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 2930  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fv 6494  df-ushgr 29039

This theorem is referenced by:  ushgrf  29043  uspgrushgr  29157