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Mirrors > Home > MPE Home > Th. List > vtxdgval | Structured version Visualization version GIF version |
Description: The degree of a vertex. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
Ref | Expression |
---|---|
vtxdgval.v | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxdgval.i | ⊢ 𝐼 = (iEdg‘𝐺) |
vtxdgval.a | ⊢ 𝐴 = dom 𝐼 |
Ref | Expression |
---|---|
vtxdgval | ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdgval.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | 1vgrex 26299 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐺 ∈ V) |
3 | vtxdgval.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | vtxdgval.a | . . . . 5 ⊢ 𝐴 = dom 𝐼 | |
5 | 1, 3, 4 | vtxdgfval 26764 | . . . 4 ⊢ (𝐺 ∈ V → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝑈 ∈ 𝑉 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))) |
7 | 6 | fveq1d 6434 | . 2 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))‘𝑈)) |
8 | eleq1 2893 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (𝐼‘𝑥) ↔ 𝑈 ∈ (𝐼‘𝑥))) | |
9 | 8 | rabbidv 3401 | . . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) |
10 | 9 | fveq2d 6436 | . . . 4 ⊢ (𝑢 = 𝑈 → (♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})) |
11 | sneq 4406 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → {𝑢} = {𝑈}) | |
12 | 11 | eqeq2d 2834 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐼‘𝑥) = {𝑢} ↔ (𝐼‘𝑥) = {𝑈})) |
13 | 12 | rabbidv 3401 | . . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) |
14 | 13 | fveq2d 6436 | . . . 4 ⊢ (𝑢 = 𝑈 → (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}) = (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})) |
15 | 10, 14 | oveq12d 6922 | . . 3 ⊢ (𝑢 = 𝑈 → ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}))) |
16 | eqid 2824 | . . 3 ⊢ (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))) | |
17 | ovex 6936 | . . 3 ⊢ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})) ∈ V | |
18 | 15, 16, 17 | fvmpt 6528 | . 2 ⊢ (𝑈 ∈ 𝑉 → ((𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}))) |
19 | 7, 18 | eqtrd 2860 | 1 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 {crab 3120 Vcvv 3413 {csn 4396 ↦ cmpt 4951 dom cdm 5341 ‘cfv 6122 (class class class)co 6904 +𝑒 cxad 12229 ♯chash 13409 Vtxcvtx 26293 iEdgciedg 26294 VtxDegcvtxdg 26762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-vtxdg 26763 |
This theorem is referenced by: vtxdgfival 26766 vtxdun 26778 vtxdlfgrval 26782 vtxd0nedgb 26785 vtxdushgrfvedg 26787 vtxdginducedm1 26840 |
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