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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 5475 | . . 3 ⊢ 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ V | |
2 | fvex 6920 | . . . . . 6 ⊢ (Vtx‘𝐺) ∈ V | |
3 | fvex 6920 | . . . . . 6 ⊢ (iEdg‘𝐺) ∈ V | |
4 | 2, 3 | opvtxfvi 29041 | . . . . 5 ⊢ (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺) |
5 | 4 | eqcomi 2744 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
6 | 2, 3 | opiedgfvi 29042 | . . . . 5 ⊢ (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺) |
7 | 6 | eqcomi 2744 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
8 | eqid 2735 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
9 | 5, 7, 8 | vtxdgfval 29500 | . . 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 7434 | . . 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 2735 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
14 | eqid 2735 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
15 | 13, 14, 8 | vtxdgfval 29500 | . 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 2786 | 1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 {csn 4631 〈cop 4637 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 +𝑒 cxad 13150 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 VtxDegcvtxdg 29498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-1st 8013 df-2nd 8014 df-vtx 29030 df-iedg 29031 df-vtxdg 29499 |
This theorem is referenced by: finsumvtxdg2size 29583 |
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