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Theorem vtxduspgrfvedgfi 16708

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 5696	. . 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 2238	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2238	. . 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 16588	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
12	10, 11	syl 14	. . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
13	4, 5, 6, 7, 8, 9, 12	vtxdgfifival 16698	. 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 16707	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
16	4, 5, 6, 7, 8, 9, 12	vtxlpfi 16697	. . . 4 ⊢ (𝜑 → {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ∈ Fin)
17		uspgrushgr 16587	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
18	10, 17	syl 14	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ USHGraph)
19		eqid 2238	. . . . . 6 ⊢ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} = {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}
20		eqeq1 2245	. . . . . . 7 ⊢ (𝑒 = 𝑐 → (𝑒 = {𝑈} ↔ 𝑐 = {𝑈}))
21	20	cbvrabv 2820	. . . . . 6 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}} = {𝑐 ∈ 𝐸 ∣ 𝑐 = {𝑈}}
22		eqid 2238	. . . . . 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 16635	. . . . 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 415	. . . 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 11244	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}}))
26	15, 25	oveq12d 6103	. 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 2275	1 ⊢ (𝜑 → (𝐷‘𝑈) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) + (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 = wceq 1402 ∈ wcel 2209 {crab 2532 {csn 3709 ↦ cmpt 4192 dom cdm 4774 –1-1-onto→wf1o 5376 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 + caddc 8183 ♯chash 11230 Vtxcvtx 16419 iEdgciedg 16420 Edgcedg 16464 USHGraphcushgr 16475 UPGraphcupgr 16498 USPGraphcuspgr 16560 VtxDegcvtxdg 16693

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-ltadd 8296

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-2o 6688 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-z 9650 df-dec 9783 df-uz 9932 df-xadd 10186 df-ihash 11231 df-ndx 13407 df-slot 13408 df-base 13410 df-edgf 16412 df-vtx 16421 df-iedg 16422 df-edg 16465 df-uhgrm 16476 df-ushgrm 16477 df-upgren 16500 df-uspgren 16562 df-vtxdg 16694

This theorem is used by: 1loopgrvd2fi 16712

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