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Mirrors > Home > MPE Home > Th. List > vtxdgop | Structured version Visualization version GIF version |
Description: The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.) |
Ref | Expression |
---|---|
vtxdgop | ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5358 | . . 3 ⊢ 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ V | |
2 | fvex 6685 | . . . . . 6 ⊢ (Vtx‘𝐺) ∈ V | |
3 | fvex 6685 | . . . . . 6 ⊢ (iEdg‘𝐺) ∈ V | |
4 | 2, 3 | opvtxfvi 26796 | . . . . 5 ⊢ (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺) |
5 | 4 | eqcomi 2832 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
6 | 2, 3 | opiedgfvi 26797 | . . . . 5 ⊢ (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺) |
7 | 6 | eqcomi 2832 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
8 | eqid 2823 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
9 | 5, 7, 8 | vtxdgfval 27251 | . . 3 ⊢ (〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ V → (VtxDeg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})))) |
10 | 1, 9 | mp1i 13 | . 2 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})))) |
11 | df-ov 7161 | . . 3 ⊢ ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)) = (VtxDeg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) | |
12 | 11 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)) = (VtxDeg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉)) |
13 | eqid 2823 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
14 | eqid 2823 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
15 | 13, 14, 8 | vtxdgfval 27251 | . 2 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ (Vtx‘𝐺) ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})))) |
16 | 10, 12, 15 | 3eqtr4rd 2869 | 1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 {csn 4569 〈cop 4575 ↦ cmpt 5148 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 +𝑒 cxad 12508 ♯chash 13693 Vtxcvtx 26783 iEdgciedg 26784 VtxDegcvtxdg 27249 |
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-rep 5192 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-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-reu 3147 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-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-1st 7691 df-2nd 7692 df-vtx 26785 df-iedg 26786 df-vtxdg 27250 |
This theorem is referenced by: finsumvtxdg2size 27334 |
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