Theorem uhgr0e 26086

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 6199	. . 3 ⊢ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
2		dm0 5446	. . . 4 ⊢ dom ∅ = ∅
3	2	feq2i 6150	. . 3 ⊢ (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	1, 3	mpbir 221	. 2 ⊢ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
5		uhgr0e.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
6		eqid 2724	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7		eqid 2724	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	isuhgr 26075	. . . 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 5431	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
13	11, 12	feq12d 6146	. . . 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 268	. 2 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
16	4, 15	mpbiri 248	1 ⊢ (𝜑 → 𝐺 ∈ UHGraph)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1596   ∈ wcel 2103   ∖ cdif 3677  ∅c0 4023  𝒫 cpw 4266  {csn 4285  dom cdm 5218  ⟶wf 5997  ‘cfv 6001  Vtxcvtx 25994  iEdgciedg 25995  UHGraphcuhgr 26071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-uhgr 26073

This theorem is referenced by:  uhgr0vb  26087