Theorem edg0iedg0 29140

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 29134	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2760	. . . 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 5894	. . . 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 6517	. . 3 ⊢ (Fun 𝐼 → Rel 𝐼)
12		relrn0 5930	. . . 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 1542  ∅c0 4287  ran crn 5633  Rel wrel 5637  Fun wfun 6494  ‘cfv 6500  iEdgciedg 29082  Edgcedg 29132

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fv 6508  df-edg 29133

This theorem is referenced by:  uhgriedg0edg0  29212  egrsubgr  29362  vtxduhgr0e  29564