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| Mirrors > Home > MPE Home > Th. List > edg0usgr | Structured version Visualization version GIF version | ||
| Description: A class without edges is a simple graph. Since ran 𝐹 = ∅ does not generally imply Fun 𝐹, but Fun (iEdg‘𝐺) is required for 𝐺 to be a simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.) |
| Ref | Expression |
|---|---|
| edg0usgr | ⊢ ((𝐺 ∈ 𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | edgval 29066 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ (𝐺 ∈ 𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
| 3 | 2 | eqeq1d 2739 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅)) |
| 4 | funrel 6583 | . . . . . 6 ⊢ (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺)) | |
| 5 | relrn0 5983 | . . . . . . 7 ⊢ (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅)) | |
| 6 | 5 | bicomd 223 | . . . . . 6 ⊢ (Rel (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
| 7 | 4, 6 | syl 17 | . . . . 5 ⊢ (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
| 8 | simpr 484 | . . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ 𝑊) | |
| 9 | simpl 482 | . . . . . . 7 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → (iEdg‘𝐺) = ∅) | |
| 10 | 8, 9 | usgr0e 29253 | . . . . . 6 ⊢ (((iEdg‘𝐺) = ∅ ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ USGraph) |
| 11 | 10 | ex 412 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → (𝐺 ∈ 𝑊 → 𝐺 ∈ USGraph)) |
| 12 | 7, 11 | biimtrdi 253 | . . . 4 ⊢ (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ → (𝐺 ∈ 𝑊 → 𝐺 ∈ USGraph))) |
| 13 | 12 | com13 88 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (ran (iEdg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph))) |
| 14 | 3, 13 | sylbid 240 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((Edg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph))) |
| 15 | 14 | 3imp 1111 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∅c0 4333 ran crn 5686 Rel wrel 5690 Fun wfun 6555 ‘cfv 6561 iEdgciedg 29014 Edgcedg 29064 USGraphcusgr 29166 |
| 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-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 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-fn 6564 df-f 6565 df-f1 6566 df-fv 6569 df-edg 29065 df-usgr 29168 |
| This theorem is referenced by: (None) |
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