Theorem edg0usgr 27037

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 26836	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	2	eqeq1d 2825	. . 3 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4		funrel 6374	. . . . . 6 ⊢ (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5		relrn0 5842	. . . . . . 7 ⊢ (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6	5	bicomd 225	. . . . . 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 487	. . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ 𝑊)
9		simpl 485	. . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → (iEdg‘𝐺) = ∅)
10	8, 9	usgr0e 27020	. . . . . 6 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ USGraph)
11	10	ex 415	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → (𝐺 ∈ 𝑊 → 𝐺 ∈ USGraph))
12	7, 11	syl6bi 255	. . . 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 242	. 2 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
15	14	3imp 1107	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∅c0 4293  ran crn 5558  Rel wrel 5562  Fun wfun 6351  ‘cfv 6357  iEdgciedg 26784  Edgcedg 26834  USGraphcusgr 26936

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fv 6365  df-edg 26835  df-usgr 26938

This theorem is referenced by: (None)