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Theorem vtxdgf 26247
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.)
Hypothesis
Ref Expression
vtxdgf.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
vtxdgf (𝐺𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ0*)

Proof of Theorem vtxdgf
Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2626 . . . . . 6 {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}
2 fvex 6160 . . . . . . 7 (iEdg‘𝐺) ∈ V
3 dmexg 7045 . . . . . . 7 ((iEdg‘𝐺) ∈ V → dom (iEdg‘𝐺) ∈ V)
42, 3mp1i 13 . . . . . 6 ((𝐺𝑊𝑢𝑉) → dom (iEdg‘𝐺) ∈ V)
51, 4rabexd 4779 . . . . 5 ((𝐺𝑊𝑢𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
6 hashxnn0 13064 . . . . 5 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) ∈ ℕ0*)
75, 6syl 17 . . . 4 ((𝐺𝑊𝑢𝑉) → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) ∈ ℕ0*)
8 eqid 2626 . . . . . 6 {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}
98, 4rabexd 4779 . . . . 5 ((𝐺𝑊𝑢𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} ∈ V)
10 hashxnn0 13064 . . . . 5 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} ∈ V → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}) ∈ ℕ0*)
119, 10syl 17 . . . 4 ((𝐺𝑊𝑢𝑉) → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}) ∈ ℕ0*)
12 xnn0xaddcl 12008 . . . 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*)
137, 11, 12syl2anc 692 . . 3 ((𝐺𝑊𝑢𝑉) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})) ∈ ℕ0*)
14 eqid 2626 . . 3 (𝑢𝑉 ↦ ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})))
1513, 14fmptd 6341 . 2 (𝐺𝑊 → (𝑢𝑉 ↦ ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))):𝑉⟶ℕ0*)
16 vtxdgf.v . . . 4 𝑉 = (Vtx‘𝐺)
17 eqid 2626 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
18 eqid 2626 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
1916, 17, 18vtxdgfval 26244 . . 3 (𝐺𝑊 → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
2019feq1d 5989 . 2 (𝐺𝑊 → ((VtxDeg‘𝐺):𝑉⟶ℕ0* ↔ (𝑢𝑉 ↦ ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))):𝑉⟶ℕ0*))
2115, 20mpbird 247 1 (𝐺𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ0*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  {crab 2916  Vcvv 3191  {csn 4153  cmpt 4678  dom cdm 5079  wf 5846  cfv 5850  (class class class)co 6605  0*cxnn0 11308   +𝑒 cxad 11888  #chash 13054  Vtxcvtx 25769  iEdgciedg 25770  VtxDegcvtxdg 26242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-xadd 11891  df-hash 13055  df-vtxdg 26243
This theorem is referenced by:  vtxdgelxnn0  26248  vtxdgfisf  26252
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