Theorem edg0iedg0 29072

Description: There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.)

Hypotheses

Ref	Expression
edg0iedg0.i	⊢ 𝐼 = (iEdg‘𝐺)
edg0iedg0.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
edg0iedg0	⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))

Proof of Theorem edg0iedg0

Step	Hyp	Ref	Expression
1		edg0iedg0.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
2		edgval 29066	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2765	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	eqeq1i 2742	. . 3 ⊢ (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)
5	4	a1i 11	. 2 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6		edg0iedg0.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
7	6	eqcomi 2746	. . . . 5 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5948	. . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	8	eqeq1i 2742	. . 3 ⊢ (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)
10	9	a1i 11	. 2 ⊢ (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11		funrel 6583	. . 3 ⊢ (Fun 𝐼 → Rel 𝐼)
12		relrn0 5983	. . . 4 ⊢ (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
13	12	bicomd 223	. . 3 ⊢ (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
14	11, 13	syl 17	. 2 ⊢ (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
15	5, 10, 14	3bitrd 305	1 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∅c0 4333  ran crn 5686  Rel wrel 5690  Fun wfun 6555  ‘cfv 6561  iEdgciedg 29014  Edgcedg 29064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-edg 29065

This theorem is referenced by:  uhgriedg0edg0  29144  egrsubgr  29294  vtxduhgr0e  29496