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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 4643 | . . . 4 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁}) | |
2 | 1 | difeq2d 4121 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁})) |
3 | preq2 4743 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁}) | |
4 | 3 | sseq1d 4011 | . . . 4 ⊢ (𝑛 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑘, 𝑁} ⊆ 𝑒)) |
5 | 4 | rexbidv 3169 | . . 3 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒)) |
6 | 2, 5 | raleqbidv 3330 | . 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 29331 | . 2 ⊢ (UnivVtx‘𝐺) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒} |
10 | 6, 9 | elrab2 3684 | 1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 ∖ cdif 3944 ⊆ wss 3947 {csn 4633 {cpr 4635 ‘cfv 6554 Vtxcvtx 28932 Edgcedg 28983 UnivVtxcuvtx 29321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-nbgr 29269 df-uvtx 29322 |
This theorem is referenced by: (None) |
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