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Theorem edg0iedg0OLD 26149
 Description: Obsolete version of edg0iedg0 26148 as of 8-Dec-2021. (Contributed by AV, 15-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
edg0iedg0.i 𝐼 = (iEdg‘𝐺)
edg0iedg0.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
edg0iedg0OLD ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))

Proof of Theorem edg0iedg0OLD
StepHypRef Expression
1 edg0iedg0.e . . . . 5 𝐸 = (Edg‘𝐺)
2 edgvalOLD 26141 . . . . 5 (𝐺𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
31, 2syl5eq 2806 . . . 4 (𝐺𝑊𝐸 = ran (iEdg‘𝐺))
43adantr 472 . . 3 ((𝐺𝑊 ∧ Fun 𝐼) → 𝐸 = ran (iEdg‘𝐺))
54eqeq1d 2762 . 2 ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6 edg0iedg0.i . . . . . 6 𝐼 = (iEdg‘𝐺)
76eqcomi 2769 . . . . 5 (iEdg‘𝐺) = 𝐼
87a1i 11 . . . 4 ((𝐺𝑊 ∧ Fun 𝐼) → (iEdg‘𝐺) = 𝐼)
98rneqd 5508 . . 3 ((𝐺𝑊 ∧ Fun 𝐼) → ran (iEdg‘𝐺) = ran 𝐼)
109eqeq1d 2762 . 2 ((𝐺𝑊 ∧ Fun 𝐼) → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11 funrel 6066 . . . 4 (Fun 𝐼 → Rel 𝐼)
12 relrn0 5538 . . . . 5 (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
1312bicomd 213 . . . 4 (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1411, 13syl 17 . . 3 (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1514adantl 473 . 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 383   = wceq 1632   ∈ wcel 2139  ∅c0 4058  ran crn 5267  Rel wrel 5271  Fun wfun 6043  ‘cfv 6049  iEdgciedg 26074  Edgcedg 26138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-edg 26139 This theorem is referenced by: (None)
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