Theorem edg0iedg0 29039

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 29033	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2759	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	eqeq1i 2741	. . 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 2745	. . . . 5 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5922	. . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	8	eqeq1i 2741	. . 3 ⊢ (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)
10	9	a1i 11	. 2 ⊢ (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11		funrel 6558	. . 3 ⊢ (Fun 𝐼 → Rel 𝐼)
12		relrn0 5957	. . . 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 4313  ran crn 5660  Rel wrel 5664  Fun wfun 6530  ‘cfv 6536  iEdgciedg 28981  Edgcedg 29031

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fv 6544  df-edg 29032

This theorem is referenced by:  uhgriedg0edg0  29111  egrsubgr  29261  vtxduhgr0e  29463