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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 5206 | . . 3 ⊢ 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ V | |
2 | fvex 6506 | . . . . . 6 ⊢ (Vtx‘𝐺) ∈ V | |
3 | fvex 6506 | . . . . . 6 ⊢ (iEdg‘𝐺) ∈ V | |
4 | 2, 3 | opvtxfvi 26487 | . . . . 5 ⊢ (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺) |
5 | 4 | eqcomi 2781 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
6 | 2, 3 | opiedgfvi 26488 | . . . . 5 ⊢ (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺) |
7 | 6 | eqcomi 2781 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
8 | eqid 2772 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
9 | 5, 7, 8 | vtxdgfval 26942 | . . 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 6973 | . . 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 2772 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
14 | eqid 2772 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
15 | 13, 14, 8 | vtxdgfval 26942 | . 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 2819 | 1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 {crab 3086 Vcvv 3409 {csn 4435 〈cop 4441 ↦ cmpt 5002 dom cdm 5400 ‘cfv 6182 (class class class)co 6970 +𝑒 cxad 12315 ♯chash 13498 Vtxcvtx 26474 iEdgciedg 26475 VtxDegcvtxdg 26940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-1st 7494 df-2nd 7495 df-vtx 26476 df-iedg 26477 df-vtxdg 26941 |
This theorem is referenced by: finsumvtxdg2size 27025 |
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