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Theorem rusgrpropedg 28838

Description: The properties of a k-regular simple graph expressed with edges. (Contributed by AV, 23-Dec-2020.) (Revised by AV, 27-Dec-2020.)

**Hypothesis**
Ref	Expression
rusgrpropnb.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
rusgrpropedg	⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))

Distinct variable groups: 𝑣,𝐺 𝑣,𝐾 𝑒,𝐺,𝑣 Allowed substitution hints: 𝐾(𝑒) 𝑉(𝑣,𝑒)

**Proof of Theorem rusgrpropedg**
Step	Hyp	Ref	Expression
1		rusgrpropnb.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	rusgrpropnb 28837	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
3		eqid 2732	. . . . . . . . 9 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	1, 3	nbedgusgr 28626	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}))
5	4	eqeq1d 2734	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ↔ (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
6	5	biimpd 228	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 → (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
7	6	ralimdva 3167	. . . . 5 ⊢ (𝐺 ∈ USGraph → (∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 → ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
8	7	adantr 481	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) → (∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 → ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
9	8	imdistani 569	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^) ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^) ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
10		df-3an 1089	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) ↔ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
11		df-3an 1089	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾) ↔ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
12	9, 10, 11	3imtr4i 291	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
13	2, 12	syl 17	1 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 ℕ₀^*cxnn0 12543 ♯chash 14289 Vtxcvtx 28253 Edgcedg 28304 USGraphcusgr 28406 NeighbVtx cnbgr 28586 RegUSGraph crusgr 28810

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-oadd 8469 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-xadd 13092 df-fz 13484 df-hash 14290 df-edg 28305 df-uhgr 28315 df-ushgr 28316 df-upgr 28339 df-umgr 28340 df-uspgr 28407 df-usgr 28408 df-nbgr 28587 df-vtxdg 28720 df-rgr 28811 df-rusgr 28812

This theorem is referenced by: (None)

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