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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 6645 | . . 3 ⊢ (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺)) | |
2 | fveq2 6645 | . . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
3 | cplgruvtxb.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 2, 3 | eqtr4di 2851 | . . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
5 | 1, 4 | eqeq12d 2814 | . 2 ⊢ (𝑔 = 𝐺 → ((UnivVtx‘𝑔) = (Vtx‘𝑔) ↔ (UnivVtx‘𝐺) = 𝑉)) |
6 | df-cplgr 27201 | . 2 ⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} | |
7 | 5, 6 | elab2g 3616 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 Vtxcvtx 26789 UnivVtxcuvtx 27175 ComplGraphccplgr 27199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-cplgr 27201 |
This theorem is referenced by: iscplgr 27205 cusgruvtxb 27212 nbcplgr 27224 |
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