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Theorem vtxdgf 29504

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 2735	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3		eqid 2735	. . 3 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4	1, 2, 3	vtxdgfval 29500	. 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 2735	. . . . 5 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)}
6		fvex 6920	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
7		dmexg 7924	. . . . . 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 5346	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑢 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
10		hashxnn0 14375	. . . 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 2735	. . . . 5 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}}
13	12, 8	rabexd 5346	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑢 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑢}} ∈ V)
14		hashxnn0 14375	. . . 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 13274	. . 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 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 {csn 4631 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℕ₀^*cxnn0 12597 +_𝑒 cxad 13150 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 VtxDegcvtxdg 29498

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-xadd 13153 df-hash 14367 df-vtxdg 29499

This theorem is referenced by: vtxdgelxnn0 29505 vtxdgfisf 29509

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