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Mirrors > Home > MPE Home > Th. List > edg0iedg0 | Structured version Visualization version GIF version |
Description: There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) |
Ref | Expression |
---|---|
edg0iedg0.i | ⊢ 𝐼 = (iEdg‘𝐺) |
edg0iedg0.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
edg0iedg0 | ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edg0iedg0.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
2 | edgaval 25794 | . . . . 5 ⊢ (𝐺 ∈ 𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺)) | |
3 | 1, 2 | syl5eq 2656 | . . . 4 ⊢ (𝐺 ∈ 𝑊 → 𝐸 = ran (iEdg‘𝐺)) |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → 𝐸 = ran (iEdg‘𝐺)) |
5 | 4 | eqeq1d 2612 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)) |
6 | edg0iedg0.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | eqcomi 2619 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (iEdg‘𝐺) = 𝐼) |
9 | 8 | rneqd 5274 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → ran (iEdg‘𝐺) = ran 𝐼) |
10 | 9 | eqeq1d 2612 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)) |
11 | funrel 5821 | . . . 4 ⊢ (Fun 𝐼 → Rel 𝐼) | |
12 | relrn0 5304 | . . . . 5 ⊢ (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅)) | |
13 | 12 | bicomd 212 | . . . 4 ⊢ (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
14 | 11, 13 | syl 17 | . . 3 ⊢ (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
15 | 14 | adantl 481 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
16 | 5, 10, 15 | 3bitrd 293 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 ran crn 5039 Rel wrel 5043 Fun wfun 5798 ‘cfv 5804 iEdgciedg 25674 Edgcedga 25792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-edga 25793 |
This theorem is referenced by: uhgriedg0edg0 25801 egrsubgr 40501 vtxduhgr0e 40693 |
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