Theorem edg0usgr 27620

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 27419	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	2	eqeq1d 2740	. . 3 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4		funrel 6451	. . . . . 6 ⊢ (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5		relrn0 5878	. . . . . . 7 ⊢ (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6	5	bicomd 222	. . . . . 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 485	. . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ 𝑊)
9		simpl 483	. . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → (iEdg‘𝐺) = ∅)
10	8, 9	usgr0e 27603	. . . . . 6 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ USGraph)
11	10	ex 413	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → (𝐺 ∈ 𝑊 → 𝐺 ∈ USGraph))
12	7, 11	syl6bi 252	. . . 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 239	. 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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∅c0 4256  ran crn 5590  Rel wrel 5594  Fun wfun 6427  ‘cfv 6433  iEdgciedg 27367  Edgcedg 27417  USGraphcusgr 27519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441  df-edg 27418  df-usgr 27521

This theorem is referenced by: (None)