Theorem edg0usgr 29229

Description: A class without edges is a simple graph. Since ran 𝐹 = ∅ does not generally imply Fun 𝐹, but Fun (iEdg‘𝐺) is required for 𝐺 to be a simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.)

Assertion

Ref	Expression
edg0usgr	⊢ ((𝐺 ∈ 𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)

Proof of Theorem edg0usgr

Step	Hyp	Ref	Expression
1		edgval 29025	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	2	eqeq1d 2733	. . 3 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4		funrel 6498	. . . . . 6 ⊢ (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5		relrn0 5912	. . . . . . 7 ⊢ (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6	5	bicomd 223	. . . . . 6 ⊢ (Rel (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
7	4, 6	syl 17	. . . . 5 ⊢ (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
8		simpr 484	. . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ 𝑊)
9		simpl 482	. . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → (iEdg‘𝐺) = ∅)
10	8, 9	usgr0e 29212	. . . . . 6 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ USGraph)
11	10	ex 412	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → (𝐺 ∈ 𝑊 → 𝐺 ∈ USGraph))
12	7, 11	biimtrdi 253	. . . 4 ⊢ (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ → (𝐺 ∈ 𝑊 → 𝐺 ∈ USGraph)))
13	12	com13 88	. . 3 ⊢ (𝐺 ∈ 𝑊 → (ran (iEdg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
14	3, 13	sylbid 240	. 2 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
15	14	3imp 1110	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∅c0 4283  ran crn 5617  Rel wrel 5621  Fun wfun 6475  ‘cfv 6481  iEdgciedg 28973  Edgcedg 29023  USGraphcusgr 29125

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

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-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fv 6489  df-edg 29024  df-usgr 29127

This theorem is referenced by: (None)