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Mirrors > Home > MPE Home > Th. List > uvtxel | Structured version Visualization version GIF version |
Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) |
Ref | Expression |
---|---|
uvtxel | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isuvtx 26016 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 UnivVertex 𝐸) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸}) | |
2 | 1 | eleq2d 2673 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})) |
3 | sneq 4135 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁}) | |
4 | 3 | difeq2d 3690 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁})) |
5 | preq2 4213 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁}) | |
6 | 5 | eleq1d 2672 | . . . 4 ⊢ (𝑛 = 𝑁 → ({𝑘, 𝑛} ∈ ran 𝐸 ↔ {𝑘, 𝑁} ∈ ran 𝐸)) |
7 | 4, 6 | raleqbidv 3129 | . . 3 ⊢ (𝑛 = 𝑁 → (∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸)) |
8 | 7 | elrab 3331 | . 2 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸} ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸)) |
9 | 2, 8 | syl6bb 275 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 {csn 4125 {cpr 4127 ran crn 5039 (class class class)co 6549 UnivVertex cuvtx 25948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-uvtx 25951 |
This theorem is referenced by: uvtxnbgra 26021 uvtxnb 26025 |
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