Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 25679 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐺 ∈ V) |
3 | vtxdgval.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | vtxdgval.a | . . . . 5 ⊢ 𝐴 = dom 𝐼 | |
5 | 1, 3, 4 | vtxdgfval 40683 | . . . 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 6105 | . 2 ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}})))‘𝑈)) |
8 | eleq1 2676 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (𝐼‘𝑥) ↔ 𝑈 ∈ (𝐼‘𝑥))) | |
9 | 8 | rabbidv 3164 | . . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) |
10 | 9 | fveq2d 6107 | . . . 4 ⊢ (𝑢 = 𝑈 → (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) = (#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})) |
11 | sneq 4135 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → {𝑢} = {𝑈}) | |
12 | 11 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐼‘𝑥) = {𝑢} ↔ (𝐼‘𝑥) = {𝑈})) |
13 | 12 | rabbidv 3164 | . . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}} = {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}}) |
14 | 13 | fveq2d 6107 | . . . 4 ⊢ (𝑢 = 𝑈 → (#‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}) = (#‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})) |
15 | 10, 14 | oveq12d 6567 | . . 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 2610 | . . 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 6577 | . . 3 ⊢ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})) ∈ V | |
18 | 15, 16, 17 | fvmpt 6191 | . 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 2644 | 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 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 {csn 4125 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 +𝑒 cxad 11820 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 VtxDegcvtxdg 40681 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 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-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-vtxdg 40682 |
This theorem is referenced by: vtxdgfival 40685 vtxdun 40696 vtxdlfgrval 40700 vtxd0nedgb 40703 vtxdushgrfvedg 40705 |
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