Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  0eusgraiff0rgracl Structured version   Visualization version   GIF version

Theorem 0eusgraiff0rgracl 26468
 Description: An undirected simple graph represented by a class is 0-regular iff it has no edges. (Contributed by AV, 3-Jan-2020.)
Assertion
Ref Expression
0eusgraiff0rgracl (𝐺 ∈ USGrph → (𝐺 RegGrph 0 ↔ (Edges‘𝐺) = ∅))

Proof of Theorem 0eusgraiff0rgracl
StepHypRef Expression
1 elusuhgra 25897 . . 3 (𝐺 ∈ USGrph → 𝐺 ∈ UHGrph )
2 uhgraopelvv 25826 . . 3 (𝐺 ∈ UHGrph → 𝐺 ∈ (V × V))
31, 2syl 17 . 2 (𝐺 ∈ USGrph → 𝐺 ∈ (V × V))
4 1st2nd2 7096 . . 3 (𝐺 ∈ (V × V) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
5 breq1 4586 . . . . . 6 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 RegGrph 0 ↔ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 0))
65adantr 480 . . . . 5 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → (𝐺 RegGrph 0 ↔ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 0))
7 eleq1 2676 . . . . . . . . 9 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ USGrph ))
8 df-br 4584 . . . . . . . . 9 ((1st𝐺) USGrph (2nd𝐺) ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ USGrph )
97, 8syl6bbr 277 . . . . . . . 8 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph ↔ (1st𝐺) USGrph (2nd𝐺)))
10 0eusgraiff0rgra 26466 . . . . . . . 8 ((1st𝐺) USGrph (2nd𝐺) → (⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 0 ↔ (2nd𝐺) = ∅))
119, 10syl6bi 242 . . . . . . 7 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph → (⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 0 ↔ (2nd𝐺) = ∅)))
1211imp 444 . . . . . 6 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → (⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 0 ↔ (2nd𝐺) = ∅))
13 usgrac 25880 . . . . . . . . . 10 (𝐺 ∈ USGrph → (1st𝐺) USGrph (2nd𝐺))
14 usgrafun 25878 . . . . . . . . . . 11 ((1st𝐺) USGrph (2nd𝐺) → Fun (2nd𝐺))
15 funrel 5821 . . . . . . . . . . 11 (Fun (2nd𝐺) → Rel (2nd𝐺))
1614, 15syl 17 . . . . . . . . . 10 ((1st𝐺) USGrph (2nd𝐺) → Rel (2nd𝐺))
1713, 16syl 17 . . . . . . . . 9 (𝐺 ∈ USGrph → Rel (2nd𝐺))
1817adantl 481 . . . . . . . 8 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → Rel (2nd𝐺))
19 relrn0 5304 . . . . . . . 8 (Rel (2nd𝐺) → ((2nd𝐺) = ∅ ↔ ran (2nd𝐺) = ∅))
2018, 19syl 17 . . . . . . 7 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → ((2nd𝐺) = ∅ ↔ ran (2nd𝐺) = ∅))
21 edgval 25868 . . . . . . . . . 10 (𝐺 ∈ USGrph → (Edges‘𝐺) = ran (2nd𝐺))
2221eqcomd 2616 . . . . . . . . 9 (𝐺 ∈ USGrph → ran (2nd𝐺) = (Edges‘𝐺))
2322adantl 481 . . . . . . . 8 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → ran (2nd𝐺) = (Edges‘𝐺))
2423eqeq1d 2612 . . . . . . 7 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → (ran (2nd𝐺) = ∅ ↔ (Edges‘𝐺) = ∅))
2520, 24bitrd 267 . . . . . 6 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → ((2nd𝐺) = ∅ ↔ (Edges‘𝐺) = ∅))
2612, 25bitrd 267 . . . . 5 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → (⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 0 ↔ (Edges‘𝐺) = ∅))
276, 26bitrd 267 . . . 4 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐺 ∈ USGrph ) → (𝐺 RegGrph 0 ↔ (Edges‘𝐺) = ∅))
2827ex 449 . . 3 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph → (𝐺 RegGrph 0 ↔ (Edges‘𝐺) = ∅)))
294, 28syl 17 . 2 (𝐺 ∈ (V × V) → (𝐺 ∈ USGrph → (𝐺 RegGrph 0 ↔ (Edges‘𝐺) = ∅)))
303, 29mpcom 37 1 (𝐺 ∈ USGrph → (𝐺 RegGrph 0 ↔ (Edges‘𝐺) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ran crn 5039  Rel wrel 5043  Fun wfun 5798  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  0cc0 9815   UHGrph cuhg 25819   USGrph cusg 25859  Edgescedg 25860   RegGrph crgra 26449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-uhgra 25821  df-umgra 25842  df-uslgra 25861  df-usgra 25862  df-edg 25865  df-vdgr 26421  df-rgra 26451 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator