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Mirrors > Home > MPE Home > Th. List > vtxdgf | Structured version Visualization version GIF version |
Description: The vertex degree function is a function from vertices to extended nonnegative integers. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
vtxdgf.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
vtxdgf | ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ0*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . . . 6 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} | |
2 | fvex 6461 | . . . . . . 7 ⊢ (iEdg‘𝐺) ∈ V | |
3 | dmexg 7377 | . . . . . . 7 ⊢ ((iEdg‘𝐺) ∈ V → dom (iEdg‘𝐺) ∈ V) | |
4 | 2, 3 | mp1i 13 | . . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → dom (iEdg‘𝐺) ∈ V) |
5 | 1, 4 | rabexd 5052 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V) |
6 | hashxnn0 13450 | . . . . 5 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) ∈ ℕ0*) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) ∈ ℕ0*) |
8 | eqid 2778 | . . . . . 6 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} | |
9 | 8, 4 | rabexd 5052 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} ∈ V) |
10 | hashxnn0 13450 | . . . . 5 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} ∈ V → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}) ∈ ℕ0*) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}) ∈ ℕ0*) |
12 | xnn0xaddcl 12383 | . . . 4 ⊢ (((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) ∈ ℕ0* ∧ (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}) ∈ ℕ0*) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})) ∈ ℕ0*) | |
13 | 7, 11, 12 | syl2anc 579 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})) ∈ ℕ0*) |
14 | 13 | fmpttd 6651 | . 2 ⊢ (𝐺 ∈ 𝑊 → (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))):𝑉⟶ℕ0*) |
15 | vtxdgf.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
16 | eqid 2778 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
17 | eqid 2778 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
18 | 15, 16, 17 | vtxdgfval 26832 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})))) |
19 | 18 | feq1d 6278 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((VtxDeg‘𝐺):𝑉⟶ℕ0* ↔ (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))):𝑉⟶ℕ0*)) |
20 | 14, 19 | mpbird 249 | 1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ0*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {crab 3094 Vcvv 3398 {csn 4398 ↦ cmpt 4967 dom cdm 5357 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℕ0*cxnn0 11719 +𝑒 cxad 12260 ♯chash 13441 Vtxcvtx 26361 iEdgciedg 26362 VtxDegcvtxdg 26830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-n0 11648 df-xnn0 11720 df-z 11734 df-uz 11998 df-xadd 12263 df-hash 13442 df-vtxdg 26831 |
This theorem is referenced by: vtxdgelxnn0 26837 vtxdgfisf 26841 |
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