Theorem usgredgleordALT 29214

Description: Alternate proof for usgredgleord 29213 based on usgriedgleord 29208. 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 6853	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
2	1	dmex 7865	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
3	2	rabex 5289	. . . 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 2729	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		usgredgleord.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
8		eqid 2729	. . . 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 3413	. . . 4 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} = {𝑓 ∈ 𝐸 ∣ 𝑁 ∈ 𝑓}
11		eqid 2729	. . . 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 29210	. . 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 14294	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
14	7, 6	usgriedgleord 29208	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≤ (♯‘𝑉))
15	13, 14	eqbrtrrd 5126	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3402  Vcvv 3444   class class class wbr 5102   ↦ cmpt 5183  dom cdm 5631  ‘cfv 6499   ≤ cle 11185  ♯chash 14271  Vtxcvtx 28976  iEdgciedg 28977  Edgcedg 29027  USGraphcusgr 29129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-xnn0 12492  df-z 12506  df-uz 12770  df-fz 13445  df-hash 14272  df-edg 29028  df-uhgr 29038  df-ushgr 29039  df-umgr 29063  df-uspgr 29130  df-usgr 29131

This theorem is referenced by: (None)