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Theorem vtxdgf 29399

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 2729	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3		eqid 2729	. . 3 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4	1, 2, 3	vtxdgfval 29395	. 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 2729	. . . . 5 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}
6		fvex 6871	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
7		dmexg 7877	. . . . . 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 5295	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
10		hashxnn0 14304	. . . 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 2729	. . . . 5 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}
13	12, 8	rabexd 5295	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} ∈ V)
14		hashxnn0 14304	. . . 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 13195	. . 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 7088	1 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3405 Vcvv 3447 {csn 4589 dom cdm 5638 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℕ₀^*cxnn0 12515 +_𝑒 cxad 13070 ♯chash 14295 Vtxcvtx 28923 iEdgciedg 28924 VtxDegcvtxdg 29393

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-xadd 13073 df-hash 14296 df-vtxdg 29394

This theorem is referenced by: vtxdgelxnn0 29400 vtxdgfisf 29404

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