MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  edg0iedg0 Structured version   Visualization version   GIF version

Theorem edg0iedg0 28583
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
StepHypRef Expression
1 edg0iedg0.e . . . . 5 𝐸 = (Edg‘𝐺)
2 edgval 28577 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
31, 2eqtri 2759 . . . 4 𝐸 = ran (iEdg‘𝐺)
43eqeq1i 2736 . . 3 (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)
54a1i 11 . 2 (Fun 𝐼 → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6 edg0iedg0.i . . . . . 6 𝐼 = (iEdg‘𝐺)
76eqcomi 2740 . . . . 5 (iEdg‘𝐺) = 𝐼
87rneqi 5936 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
98eqeq1i 2736 . . 3 (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)
109a1i 11 . 2 (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11 funrel 6565 . . 3 (Fun 𝐼 → Rel 𝐼)
12 relrn0 5968 . . . 4 (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
1312bicomd 222 . . 3 (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1411, 13syl 17 . 2 (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
155, 10, 143bitrd 305 1 (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  c0 4322  ran crn 5677  Rel wrel 5681  Fun wfun 6537  cfv 6543  iEdgciedg 28525  Edgcedg 28575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-edg 28576
This theorem is referenced by:  uhgriedg0edg0  28655  egrsubgr  28802  vtxduhgr0e  29003
  Copyright terms: Public domain W3C validator