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| Mirrors > Home > MPE Home > Th. List > cplgruvtxb | Structured version Visualization version GIF version | ||
| Description: A graph 𝐺 is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 15-Feb-2022.) |
| Ref | Expression |
|---|---|
| cplgruvtxb.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| Ref | Expression |
|---|---|
| cplgruvtxb | ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . 3 ⊢ (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺)) | |
| 2 | fveq2 6906 | . . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
| 3 | cplgruvtxb.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 4 | 2, 3 | eqtr4di 2795 | . . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
| 5 | 1, 4 | eqeq12d 2753 | . 2 ⊢ (𝑔 = 𝐺 → ((UnivVtx‘𝑔) = (Vtx‘𝑔) ↔ (UnivVtx‘𝐺) = 𝑉)) |
| 6 | df-cplgr 29428 | . 2 ⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} | |
| 7 | 5, 6 | elab2g 3680 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Vtxcvtx 29013 UnivVtxcuvtx 29402 ComplGraphccplgr 29426 |
| 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-ext 2708 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-iota 6514 df-fv 6569 df-cplgr 29428 |
| This theorem is referenced by: iscplgr 29432 cusgruvtxb 29439 nbcplgr 29451 |
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