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Theorem vtxduspgrfvedgfi 16422

Description: The value of the vertex degree function for a simple pseudograph. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)

**Hypotheses**
Ref	Expression
vtxdushgrfvedg.v	⊢ 𝑉 = (Vtx‘𝐺)
vtxdushgrfvedg.e	⊢ 𝐸 = (Edg‘𝐺)
vtxduspgrfvedgfi.fi	⊢ (𝜑 → dom (iEdg‘𝐺) ∈ Fin)
vtxduspgrfvedgfi.v	⊢ (𝜑 → 𝑉 ∈ Fin)
vtxduspgrfvedgfi.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
vtxduspgrfvedgfi.g	⊢ (𝜑 → 𝐺 ∈ USPGraph)
vtxdushgrfvedg.d	⊢ 𝐷 = (VtxDeg‘𝐺)

**Assertion**
Ref	Expression
vtxduspgrfvedgfi	⊢ (𝜑 → (𝐷‘𝑈) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) + (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})))

Distinct variable groups: 𝑒,𝐸 𝑒,𝐺 𝑈,𝑒 𝑒,𝑉 Allowed substitution hints: 𝜑(𝑒) 𝐷(𝑒)

Proof of Theorem vtxduspgrfvedgfi Dummy variables 𝑐 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdushgrfvedg.d	. . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺)
2	1	fveq1i 5676	. . 3 ⊢ (𝐷‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)
3	2	a1i 9	. 2 ⊢ (𝜑 → (𝐷‘𝑈) = ((VtxDeg‘𝐺)‘𝑈))
4		vtxdushgrfvedg.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
5		eqid 2234	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2234	. . 3 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
7		vtxduspgrfvedgfi.fi	. . 3 ⊢ (𝜑 → dom (iEdg‘𝐺) ∈ Fin)
8		vtxduspgrfvedgfi.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ Fin)
9		vtxduspgrfvedgfi.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
10		vtxduspgrfvedgfi.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ USPGraph)
11		uspgrupgr 16302	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
12	10, 11	syl 14	. . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
13	4, 5, 6, 7, 8, 9, 12	vtxdgfifival 16412	. 2 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) + (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}})))
14		vtxdushgrfvedg.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
15	4, 14, 7, 8, 9, 10	vtxduspgrfvedgfilem 16421	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
16	4, 5, 6, 7, 8, 9, 12	vtxlpfi 16411	. . . 4 ⊢ (𝜑 → {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ∈ Fin)
17		uspgrushgr 16301	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
18	10, 17	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ USHGraph)
19		eqid 2234	. . . . . 6 ⊢ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} = {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}
20		eqeq1 2241	. . . . . . 7 ⊢ (𝑒 = 𝑐 → (𝑒 = {𝑈} ↔ 𝑐 = {𝑈}))
21	20	cbvrabv 2814	. . . . . 6 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}} = {𝑐 ∈ 𝐸 ∣ 𝑐 = {𝑈}}
22		eqid 2234	. . . . . 6 ⊢ (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ↦ ((iEdg‘𝐺)‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ↦ ((iEdg‘𝐺)‘𝑥))
23	14, 5, 19, 21, 22	ushgredgedgloop 16349	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ↦ ((iEdg‘𝐺)‘𝑥)):{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})
24	18, 9, 23	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ↦ ((iEdg‘𝐺)‘𝑥)):{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})
25	16, 24	fihasheqf1od 11177	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}}))
26	15, 25	oveq12d 6076	. 2 ⊢ (𝜑 → ((♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) + (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}})) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) + (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})))
27	3, 13, 26	3eqtrd 2271	1 ⊢ (𝜑 → (𝐷‘𝑈) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) + (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1398 ∈ wcel 2205 {crab 2526 {csn 3694 ↦ cmpt 4176 dom cdm 4754 –1-1-onto→wf1o 5356 ‘cfv 5357 (class class class)co 6058 Fincfn 6988 + caddc 8146 ♯chash 11163 Vtxcvtx 16133 iEdgciedg 16134 Edgcedg 16178 USHGraphcushgr 16189 UPGraphcupgr 16212 USPGraphcuspgr 16274 VtxDegcvtxdg 16407

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-ltadd 8259

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-2o 6661 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-7 9318 df-8 9319 df-9 9320 df-n0 9514 df-z 9595 df-dec 9728 df-uz 9872 df-xadd 10125 df-ihash 11164 df-ndx 13299 df-slot 13300 df-base 13302 df-edgf 16126 df-vtx 16135 df-iedg 16136 df-edg 16179 df-uhgrm 16190 df-ushgrm 16191 df-upgren 16214 df-uspgren 16276 df-vtxdg 16408

This theorem is referenced by: 1loopgrvd2fi 16426

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