Theorem edg0iedg0 28180

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 28174	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2759	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	eqeq1i 2736	. . 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 2740	. . . . 5 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5928	. . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	8	eqeq1i 2736	. . 3 ⊢ (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)
10	9	a1i 11	. 2 ⊢ (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11		funrel 6554	. . 3 ⊢ (Fun 𝐼 → Rel 𝐼)
12		relrn0 5960	. . . 4 ⊢ (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
13	12	bicomd 222	. . 3 ⊢ (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
14	11, 13	syl 17	. 2 ⊢ (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
15	5, 10, 14	3bitrd 304	1 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541  ∅c0 4318  ran crn 5670  Rel wrel 5674  Fun wfun 6526  ‘cfv 6532  iEdgciedg 28122  Edgcedg 28172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fv 6540  df-edg 28173

This theorem is referenced by:  uhgriedg0edg0  28252  egrsubgr  28399  vtxduhgr0e  28600