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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 6896 | . . 3 ⊢ (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺)) | |
2 | fveq2 6896 | . . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
3 | cplgruvtxb.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 2, 3 | eqtr4di 2783 | . . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
5 | 1, 4 | eqeq12d 2741 | . 2 ⊢ (𝑔 = 𝐺 → ((UnivVtx‘𝑔) = (Vtx‘𝑔) ↔ (UnivVtx‘𝐺) = 𝑉)) |
6 | df-cplgr 29296 | . 2 ⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} | |
7 | 5, 6 | elab2g 3666 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 Vtxcvtx 28881 UnivVtxcuvtx 29270 ComplGraphccplgr 29294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-cplgr 29296 |
This theorem is referenced by: iscplgr 29300 cusgruvtxb 29307 nbcplgr 29319 |
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