Theorem uhgr0e 29165

Description: The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)

Hypotheses

Ref	Expression
uhgr0e.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
uhgr0e.e	⊢ (𝜑 → (iEdg‘𝐺) = ∅)

Assertion

Ref	Expression
uhgr0e	⊢ (𝜑 → 𝐺 ∈ UHGraph)

Proof of Theorem uhgr0e

Step	Hyp	Ref	Expression
1		f0 6715	. . 3 ⊢ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
2		dm0 5869	. . . 4 ⊢ dom ∅ = ∅
3	2	feq2i 6654	. . 3 ⊢ (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	1, 3	mpbir 232	. 2 ⊢ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
5		uhgr0e.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
6		eqid 2740	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7		eqid 2740	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	isuhgr 29154	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
9	5, 8	syl 17	. . 3 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
10		uhgr0e.e	. . . 4 ⊢ (𝜑 → (iEdg‘𝐺) = ∅)
11		id 22	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → (iEdg‘𝐺) = ∅)
12		dmeq 5852	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
13	11, 12	feq12d 6650	. . . 4 ⊢ ((iEdg‘𝐺) = ∅ → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
14	10, 13	syl 17	. . 3 ⊢ (𝜑 → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
15	9, 14	bitrd 280	. 2 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
16	4, 15	mpbiri 259	1 ⊢ (𝜑 → 𝐺 ∈ UHGraph)

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119   ∖ cdif 3887  ∅c0 4268  𝒫 cpw 4536  {csn 4562  dom cdm 5625  ⟶wf 6488  ‘cfv 6492  Vtxcvtx 29090  iEdgciedg 29091  UHGraphcuhgr 29150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-uhgr 29152

This theorem is referenced by:  uhgr0vb  29166