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Theorem vtxdgf 29489

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	⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)

Proof of Theorem vtxdgf Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdgf.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2737	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3		eqid 2737	. . 3 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4	1, 2, 3	vtxdgfval 29485	. 2 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}))))
5		eqid 2737	. . . . 5 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}
6		fvex 6919	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
7		dmexg 7923	. . . . . 6 ⊢ ((iEdg‘𝐺) ∈ V → dom (iEdg‘𝐺) ∈ V)
8	6, 7	mp1i 13	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → dom (iEdg‘𝐺) ∈ V)
9	5, 8	rabexd 5340	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
10		hashxnn0 14378	. . . 4 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) ∈ ℕ₀^*)
11	9, 10	syl 17	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) ∈ ℕ₀^*)
12		eqid 2737	. . . . 5 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}
13	12, 8	rabexd 5340	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} ∈ V)
14		hashxnn0 14378	. . . 4 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} ∈ V → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}) ∈ ℕ₀^*)
15	13, 14	syl 17	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}) ∈ ℕ₀^*)
16		xnn0xaddcl 13277	. . 3 ⊢ (((♯‘{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^)
17	11, 15, 16	syl2anc 584	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}})) ∈ ℕ₀^*)
18	4, 17	fmpt3d 7136	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 {csn 4626 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℕ₀^*cxnn0 12599 +_𝑒 cxad 13152 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 VtxDegcvtxdg 29483

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-hash 14370 df-vtxdg 29484

This theorem is referenced by: vtxdgelxnn0 29490 vtxdgfisf 29494

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