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Theorem vtxduspgrfvedgfi 16747

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 5699	. . 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 16627	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
12	10, 11	syl 14	. . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
13	4, 5, 6, 7, 8, 9, 12	vtxdgfifival 16737	. 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 16746	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
16	4, 5, 6, 7, 8, 9, 12	vtxlpfi 16736	. . . 4 ⊢ (𝜑 → {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}} ∈ Fin)
17		uspgrushgr 16626	. . . . . 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 16674	. . . . 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 11250	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) = {𝑈}}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}}))
26	15, 25	oveq12d 6106	. 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 4193 dom cdm 4776 –1-1-onto→wf1o 5379 ‘cfv 5380 (class class class)co 6088 Fincfn 7026 + caddc 8188 ♯chash 11236 Vtxcvtx 16458 iEdgciedg 16459 Edgcedg 16503 USHGraphcushgr 16514 UPGraphcupgr 16537 USPGraphcuspgr 16599 VtxDegcvtxdg 16732

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-ltadd 8301

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-1o 6691 df-2o 6692 df-er 6811 df-en 7027 df-dom 7028 df-fin 7029 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-5 9375 df-6 9376 df-7 9377 df-8 9378 df-9 9379 df-n0 9575 df-z 9656 df-dec 9789 df-uz 9938 df-xadd 10192 df-ihash 11237 df-ndx 13413 df-slot 13414 df-base 13416 df-edgf 16451 df-vtx 16460 df-iedg 16461 df-edg 16504 df-uhgrm 16515 df-ushgrm 16516 df-upgren 16539 df-uspgren 16601 df-vtxdg 16733

This theorem is used by: 1loopgrvd2fi 16751

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