Theorem usgredgleordALT 29212

Description: Alternate proof for usgredgleord 29211 based on usgriedgleord 29206. 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 6835	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
2	1	dmex 7839	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
3	2	rabex 5275	. . . 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 2731	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		usgredgleord.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
8		eqid 2731	. . . 4 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}
9		eleq2w 2815	. . . . 5 ⊢ (𝑒 = 𝑓 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑓))
10	9	cbvrabv 3405	. . . 4 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} = {𝑓 ∈ 𝐸 ∣ 𝑁 ∈ 𝑓}
11		eqid 2731	. . . 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 29208	. . 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 14260	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
14	7, 6	usgriedgleord 29206	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≤ (♯‘𝑉))
15	13, 14	eqbrtrrd 5113	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   class class class wbr 5089   ↦ cmpt 5170  dom cdm 5614  ‘cfv 6481   ≤ cle 11147  ♯chash 14237  Vtxcvtx 28974  iEdgciedg 28975  Edgcedg 29025  USGraphcusgr 29127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-xnn0 12455  df-z 12469  df-uz 12733  df-fz 13408  df-hash 14238  df-edg 29026  df-uhgr 29036  df-ushgr 29037  df-umgr 29061  df-uspgr 29128  df-usgr 29129

This theorem is referenced by: (None)