Theorem edg0iedg0OLD 25950

Description: Obsolete version of edg0iedg0 25949 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	⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))

Proof of Theorem edg0iedg0OLD

Step	Hyp	Ref	Expression
1		edg0iedg0.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
2		edgvalOLD 25942	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	1, 2	syl5eq 2668	. . . 4 ⊢ (𝐺 ∈ 𝑊 → 𝐸 = ran (iEdg‘𝐺))
4	3	adantr 481	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → 𝐸 = ran (iEdg‘𝐺))
5	4	eqeq1d 2624	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6		edg0iedg0.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
7	6	eqcomi 2631	. . . . 5 ⊢ (iEdg‘𝐺) = 𝐼
8	7	a1i 11	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (iEdg‘𝐺) = 𝐼)
9	8	rneqd 5353	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → ran (iEdg‘𝐺) = ran 𝐼)
10	9	eqeq1d 2624	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11		funrel 5905	. . . 4 ⊢ (Fun 𝐼 → Rel 𝐼)
12		relrn0 5383	. . . . 5 ⊢ (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
13	12	bicomd 213	. . . 4 ⊢ (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
14	11, 13	syl 17	. . 3 ⊢ (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
15	14	adantl 482	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
16	5, 10, 15	3bitrd 294	1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∅c0 3915  ran crn 5115  Rel wrel 5119  Fun wfun 5882  ‘cfv 5888  iEdgciedg 25875  Edgcedg 25939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-edg 25940

This theorem is referenced by: (None)