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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.) (Revised by AV, 8-Dec-2021.) |
Ref | Expression |
---|---|
edg0iedg0.i | ⊢ 𝐼 = (iEdg‘𝐺) |
edg0iedg0.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
edg0iedg0 | ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edg0iedg0.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
2 | edgval 29081 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
3 | 1, 2 | eqtri 2763 | . . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺) |
4 | 3 | eqeq1i 2740 | . . 3 ⊢ (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅) |
5 | 4 | a1i 11 | . 2 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)) |
6 | edg0iedg0.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | eqcomi 2744 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | rneqi 5951 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
9 | 8 | eqeq1i 2740 | . . 3 ⊢ (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅) |
10 | 9 | a1i 11 | . 2 ⊢ (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)) |
11 | funrel 6585 | . . 3 ⊢ (Fun 𝐼 → Rel 𝐼) | |
12 | relrn0 5986 | . . . 4 ⊢ (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅)) | |
13 | 12 | bicomd 223 | . . 3 ⊢ (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
14 | 11, 13 | syl 17 | . 2 ⊢ (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
15 | 5, 10, 14 | 3bitrd 305 | 1 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∅c0 4339 ran crn 5690 Rel wrel 5694 Fun wfun 6557 ‘cfv 6563 iEdgciedg 29029 Edgcedg 29079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-edg 29080 |
This theorem is referenced by: uhgriedg0edg0 29159 egrsubgr 29309 vtxduhgr0e 29511 |
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