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Mirrors > Home > MPE Home > Th. List > uvtxel1 | Structured version Visualization version GIF version |
Description: Characterization of a universal vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 14-Feb-2022.) |
Ref | Expression |
---|---|
uvtxel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isuvtx.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
uvtxel1 | ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4579 | . . . 4 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁}) | |
2 | 1 | difeq2d 4101 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁})) |
3 | preq2 4672 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁}) | |
4 | 3 | sseq1d 4000 | . . . 4 ⊢ (𝑛 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑘, 𝑁} ⊆ 𝑒)) |
5 | 4 | rexbidv 3299 | . . 3 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒)) |
6 | 2, 5 | raleqbidv 3403 | . 2 ⊢ (𝑛 = 𝑁 → (∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒)) |
7 | uvtxel.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | isuvtx.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
9 | 7, 8 | isuvtx 27179 | . 2 ⊢ (UnivVtx‘𝐺) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒} |
10 | 6, 9 | elrab2 3685 | 1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 {cpr 4571 ‘cfv 6357 Vtxcvtx 26783 Edgcedg 26834 UnivVtxcuvtx 27169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-nbgr 27117 df-uvtx 27170 |
This theorem is referenced by: (None) |
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