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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 29066 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
| 3 | 1, 2 | eqtri 2765 | . . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺) | 
| 4 | 3 | eqeq1i 2742 | . . 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 2746 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 | 
| 8 | 7 | rneqi 5948 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 | 
| 9 | 8 | eqeq1i 2742 | . . 3 ⊢ (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅) | 
| 10 | 9 | a1i 11 | . 2 ⊢ (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)) | 
| 11 | funrel 6583 | . . 3 ⊢ (Fun 𝐼 → Rel 𝐼) | |
| 12 | relrn0 5983 | . . . 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 1540 ∅c0 4333 ran crn 5686 Rel wrel 5690 Fun wfun 6555 ‘cfv 6561 iEdgciedg 29014 Edgcedg 29064 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-edg 29065 | 
| This theorem is referenced by: uhgriedg0edg0 29144 egrsubgr 29294 vtxduhgr0e 29496 | 
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