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Theorem vtxduspgrfvedgfi 16225

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 5649	. . 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 2231	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2231	. . 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 16105	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
12	10, 11	syl 14	. . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
13	4, 5, 6, 7, 8, 9, 12	vtxdgfifival 16215	. 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 16224	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
16	4, 5, 6, 7, 8, 9, 12	vtxlpfi 16214	. . . 4 ⊢ (𝜑 → {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ∈ Fin)
17		uspgrushgr 16104	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
18	10, 17	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ USHGraph)
19		eqid 2231	. . . . . 6 ⊢ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} = {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}
20		eqeq1 2238	. . . . . . 7 ⊢ (𝑒 = 𝑐 → (𝑒 = {𝑈} ↔ 𝑐 = {𝑈}))
21	20	cbvrabv 2802	. . . . . 6 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}} = {𝑐 ∈ 𝐸 ∣ 𝑐 = {𝑈}}
22		eqid 2231	. . . . . 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 16152	. . . . 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 11097	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}}))
26	15, 25	oveq12d 6046	. 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 2268	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 2202 {crab 2515 {csn 3673 ↦ cmpt 4155 dom cdm 4731 –1-1-onto→wf1o 5332 ‘cfv 5333 (class class class)co 6028 Fincfn 6952 + caddc 8078 ♯chash 11083 Vtxcvtx 15936 iEdgciedg 15937 Edgcedg 15981 USHGraphcushgr 15992 UPGraphcupgr 16015 USPGraphcuspgr 16077 VtxDegcvtxdg 16210

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-ltadd 8191

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-1o 6625 df-2o 6626 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-5 9247 df-6 9248 df-7 9249 df-8 9250 df-9 9251 df-n0 9445 df-z 9524 df-dec 9656 df-uz 9800 df-xadd 10052 df-ihash 11084 df-ndx 13148 df-slot 13149 df-base 13151 df-edgf 15929 df-vtx 15938 df-iedg 15939 df-edg 15982 df-uhgrm 15993 df-ushgrm 15994 df-upgren 16017 df-uspgren 16079 df-vtxdg 16211

This theorem is referenced by: 1loopgrvd2fi 16229

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