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Theorem edg0iedg0 25880
Description: There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.)
Hypotheses
Ref Expression
edg0iedg0.i 𝐼 = (iEdg‘𝐺)
edg0iedg0.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
edg0iedg0 ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))

Proof of Theorem edg0iedg0
StepHypRef Expression
1 edg0iedg0.e . . . . 5 𝐸 = (Edg‘𝐺)
2 edgval 25875 . . . . 5 (𝐺𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
31, 2syl5eq 2667 . . . 4 (𝐺𝑊𝐸 = ran (iEdg‘𝐺))
43adantr 481 . . 3 ((𝐺𝑊 ∧ Fun 𝐼) → 𝐸 = ran (iEdg‘𝐺))
54eqeq1d 2623 . 2 ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6 edg0iedg0.i . . . . . 6 𝐼 = (iEdg‘𝐺)
76eqcomi 2630 . . . . 5 (iEdg‘𝐺) = 𝐼
87a1i 11 . . . 4 ((𝐺𝑊 ∧ Fun 𝐼) → (iEdg‘𝐺) = 𝐼)
98rneqd 5323 . . 3 ((𝐺𝑊 ∧ Fun 𝐼) → ran (iEdg‘𝐺) = ran 𝐼)
109eqeq1d 2623 . 2 ((𝐺𝑊 ∧ Fun 𝐼) → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11 funrel 5874 . . . 4 (Fun 𝐼 → Rel 𝐼)
12 relrn0 5353 . . . . 5 (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
1312bicomd 213 . . . 4 (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1411, 13syl 17 . . 3 (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1514adantl 482 . 2 ((𝐺𝑊 ∧ Fun 𝐼) → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
165, 10, 153bitrd 294 1 ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  c0 3897  ran crn 5085  Rel wrel 5089  Fun wfun 5851  cfv 5857  iEdgciedg 25809  Edgcedg 25873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fv 5865  df-edg 25874
This theorem is referenced by:  uhgriedg0edg0  25951  egrsubgr  26096  vtxduhgr0e  26294
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