Theorem edg0usgr 26550

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 26347	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	2	eqeq1d 2827	. . 3 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4		funrel 6140	. . . . . 6 ⊢ (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5		relrn0 5616	. . . . . . 7 ⊢ (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6	5	bicomd 215	. . . . . 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 479	. . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ 𝑊)
9		simpl 476	. . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → (iEdg‘𝐺) = ∅)
10	8, 9	usgr0e 26533	. . . . . 6 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ USGraph)
11	10	ex 403	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → (𝐺 ∈ 𝑊 → 𝐺 ∈ USGraph))
12	7, 11	syl6bi 245	. . . 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 232	. 2 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
15	14	3imp 1141	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∅c0 4144  ran crn 5343  Rel wrel 5347  Fun wfun 6117  ‘cfv 6123  iEdgciedg 26295  Edgcedg 26345  USGraphcusgr 26448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fv 6131  df-edg 26346  df-usgr 26450

This theorem is referenced by: (None)