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Mirrors > Home > MPE Home > Th. List > vtxdumgrval | Structured version Visualization version GIF version |
Description: The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 23-Feb-2021.) |
Ref | Expression |
---|---|
vtxdlfgrval.v | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxdlfgrval.i | ⊢ 𝐼 = (iEdg‘𝐺) |
vtxdlfgrval.a | ⊢ 𝐴 = dom 𝐼 |
vtxdlfgrval.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
vtxdumgrval | ⊢ ((𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdlfgrval.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | vtxdlfgrval.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | umgrislfupgr 27521 | . . 3 ⊢ (𝐺 ∈ UMGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})) |
4 | vtxdlfgrval.a | . . . . . 6 ⊢ 𝐴 = dom 𝐼 | |
5 | 4 | eqcomi 2742 | . . . . 5 ⊢ dom 𝐼 = 𝐴 |
6 | 5 | feq2i 6610 | . . . 4 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
7 | 6 | biimpi 215 | . . 3 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
8 | 3, 7 | simplbiim 504 | . 2 ⊢ (𝐺 ∈ UMGraph → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
9 | vtxdlfgrval.d | . . 3 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
10 | 1, 2, 4, 9 | vtxdlfgrval 27880 | . 2 ⊢ ((𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})) |
11 | 8, 10 | sylan 579 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 {crab 3221 𝒫 cpw 4536 class class class wbr 5077 dom cdm 5591 ⟶wf 6443 ‘cfv 6447 ≤ cle 11038 2c2 12056 ♯chash 14072 Vtxcvtx 27394 iEdgciedg 27395 UPGraphcupgr 27478 UMGraphcumgr 27479 VtxDegcvtxdg 27860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-n0 12262 df-xnn0 12334 df-z 12348 df-uz 12611 df-xadd 12877 df-fz 13268 df-hash 14073 df-upgr 27480 df-umgr 27481 df-vtxdg 27861 |
This theorem is referenced by: vtxdusgrval 27882 umgr2v2evd2 27922 vdn0conngrumgrv2 28588 |
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