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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 5454 | . . 3 ⊢ 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ V | |
2 | fvex 6894 | . . . . . 6 ⊢ (Vtx‘𝐺) ∈ V | |
3 | fvex 6894 | . . . . . 6 ⊢ (iEdg‘𝐺) ∈ V | |
4 | 2, 3 | opvtxfvi 28693 | . . . . 5 ⊢ (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺) |
5 | 4 | eqcomi 2733 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
6 | 2, 3 | opiedgfvi 28694 | . . . . 5 ⊢ (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺) |
7 | 6 | eqcomi 2733 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
8 | eqid 2724 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
9 | 5, 7, 8 | vtxdgfval 29148 | . . 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 7404 | . . 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 2724 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
14 | eqid 2724 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
15 | 13, 14, 8 | vtxdgfval 29148 | . 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 2775 | 1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 {csn 4620 〈cop 4626 ↦ cmpt 5221 dom cdm 5666 ‘cfv 6533 (class class class)co 7401 +𝑒 cxad 13086 ♯chash 14286 Vtxcvtx 28680 iEdgciedg 28681 VtxDegcvtxdg 29146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-1st 7968 df-2nd 7969 df-vtx 28682 df-iedg 28683 df-vtxdg 29147 |
This theorem is referenced by: finsumvtxdg2size 29231 |
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