Theorem usgredgleordALT 29103

Description: Alternate proof for usgredgleord 29102 based on usgriedgleord 29097. In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) (Proof shortened by AV, 5-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
usgredgleord.v	⊢ 𝑉 = (Vtx‘𝐺)
usgredgleord.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
usgredgleordALT	⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉))

Distinct variable groups:   𝑒,𝐸   𝑒,𝑁

Allowed substitution hints:   𝐺(𝑒)   𝑉(𝑒)

Proof of Theorem usgredgleordALT

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6907	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
2	1	dmex 7915	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
3	2	rabex 5334	. . . 4 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V
4	3	a1i 11	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
5		usgredgleord.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
6		eqid 2725	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		usgredgleord.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
8		eqid 2725	. . . 4 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}
9		eleq2w 2809	. . . . 5 ⊢ (𝑒 = 𝑓 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑓))
10	9	cbvrabv 3430	. . . 4 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} = {𝑓 ∈ 𝐸 ∣ 𝑁 ∈ 𝑓}
11		eqid 2725	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ↦ ((iEdg‘𝐺)‘𝑦)) = (𝑦 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ↦ ((iEdg‘𝐺)‘𝑦))
12	5, 6, 7, 8, 10, 11	usgredgedg 29099	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ↦ ((iEdg‘𝐺)‘𝑦)):{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒})
13	4, 12	hasheqf1od 14344	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
14	7, 6	usgriedgleord 29097	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≤ (♯‘𝑉))
15	13, 14	eqbrtrrd 5172	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3419  Vcvv 3463   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5677  ‘cfv 6547   ≤ cle 11279  ♯chash 14321  Vtxcvtx 28865  iEdgciedg 28866  Edgcedg 28916  USGraphcusgr 29018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-n0 12503  df-xnn0 12575  df-z 12589  df-uz 12853  df-fz 13517  df-hash 14322  df-edg 28917  df-uhgr 28927  df-ushgr 28928  df-umgr 28952  df-uspgr 29019  df-usgr 29020

This theorem is referenced by: (None)