Theorem usgredgleordALT 29034

Description: Alternate proof for usgredgleord 29033 based on usgriedgleord 29028. 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 6904	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
2	1	dmex 7911	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
3	2	rabex 5328	. . . 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 2727	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		usgredgleord.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
8		eqid 2727	. . . 4 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}
9		eleq2w 2812	. . . . 5 ⊢ (𝑒 = 𝑓 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑓))
10	9	cbvrabv 3437	. . . 4 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} = {𝑓 ∈ 𝐸 ∣ 𝑁 ∈ 𝑓}
11		eqid 2727	. . . 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 29030	. . 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 14336	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
14	7, 6	usgriedgleord 29028	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≤ (♯‘𝑉))
15	13, 14	eqbrtrrd 5166	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3427  Vcvv 3469   class class class wbr 5142   ↦ cmpt 5225  dom cdm 5672  ‘cfv 6542   ≤ cle 11271  ♯chash 14313  Vtxcvtx 28796  iEdgciedg 28797  Edgcedg 28847  USGraphcusgr 28949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-fz 13509  df-hash 14314  df-edg 28848  df-uhgr 28858  df-ushgr 28859  df-umgr 28883  df-uspgr 28950  df-usgr 28951

This theorem is referenced by: (None)