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Theorem vtxduspgrfvedgfi 16107

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 5636	. . 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 2229	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2229	. . 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 16020	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
12	10, 11	syl 14	. . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
13	4, 5, 6, 7, 8, 9, 12	vtxdgfifival 16097	. 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 16106	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
16	4, 5, 6, 7, 8, 9, 12	vtxlpfi 16096	. . . 4 ⊢ (𝜑 → {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ∈ Fin)
17		uspgrushgr 16019	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
18	10, 17	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ USHGraph)
19		eqid 2229	. . . . . 6 ⊢ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} = {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}
20		eqeq1 2236	. . . . . . 7 ⊢ (𝑒 = 𝑐 → (𝑒 = {𝑈} ↔ 𝑐 = {𝑈}))
21	20	cbvrabv 2799	. . . . . 6 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}} = {𝑐 ∈ 𝐸 ∣ 𝑐 = {𝑈}}
22		eqid 2229	. . . . . 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 16067	. . . . 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 11041	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}}))
26	15, 25	oveq12d 6031	. 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 2266	1 ⊢ (𝜑 → (𝐷‘𝑈) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) + (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1395 ∈ wcel 2200 {crab 2512 {csn 3667 ↦ cmpt 4148 dom cdm 4723 –1-1-onto→wf1o 5323 ‘cfv 5324 (class class class)co 6013 Fincfn 6904 + caddc 8025 ♯chash 11027 Vtxcvtx 15853 iEdgciedg 15854 Edgcedg 15898 USHGraphcushgr 15909 UPGraphcupgr 15932 USPGraphcuspgr 15992 VtxDegcvtxdg 16092

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-ltadd 8138

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-1o 6577 df-2o 6578 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-5 9195 df-6 9196 df-7 9197 df-8 9198 df-9 9199 df-n0 9393 df-z 9470 df-dec 9602 df-uz 9746 df-xadd 9998 df-ihash 11028 df-ndx 13075 df-slot 13076 df-base 13078 df-edgf 15846 df-vtx 15855 df-iedg 15856 df-edg 15899 df-uhgrm 15910 df-ushgrm 15911 df-upgren 15934 df-uspgren 15994 df-vtxdg 16093

This theorem is referenced by: 1loopgrvd2fi 16111

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