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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 4932 | . . 3 ⊢ 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ V | |
2 | fvex 6201 | . . . . . 6 ⊢ (Vtx‘𝐺) ∈ V | |
3 | fvex 6201 | . . . . . 6 ⊢ (iEdg‘𝐺) ∈ V | |
4 | 2, 3 | opvtxfvi 25889 | . . . . 5 ⊢ (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺) |
5 | 4 | eqcomi 2631 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
6 | 2, 3 | opiedgfvi 25890 | . . . . 5 ⊢ (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺) |
7 | 6 | eqcomi 2631 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
8 | eqid 2622 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
9 | 5, 7, 8 | vtxdgfval 26363 | . . 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 6653 | . . 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 2622 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
14 | eqid 2622 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
15 | 13, 14, 8 | vtxdgfval 26363 | . 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 2667 | 1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 {csn 4177 〈cop 4183 ↦ cmpt 4729 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 +𝑒 cxad 11944 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 VtxDegcvtxdg 26361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-1st 7168 df-2nd 7169 df-vtx 25876 df-iedg 25877 df-vtxdg 26362 |
This theorem is referenced by: finsumvtxdg2size 26446 |
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