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

Theorem sizusglecusg 26415
 Description: The size of a simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
Hypotheses
Ref Expression
fusgrmaxsize.v 𝑉 = (Vtx‘𝐺)
fusgrmaxsize.e 𝐸 = (Edg‘𝐺)
usgrsscusgra.h 𝑉 = (Vtx‘𝐻)
usgrsscusgra.f 𝐹 = (Edg‘𝐻)
Assertion
Ref Expression
sizusglecusg ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘𝐹))

Proof of Theorem sizusglecusg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fusgrmaxsize.e . . . . . . . . 9 𝐸 = (Edg‘𝐺)
2 fvex 6239 . . . . . . . . 9 (Edg‘𝐺) ∈ V
31, 2eqeltri 2726 . . . . . . . 8 𝐸 ∈ V
4 resiexg 7144 . . . . . . . 8 (𝐸 ∈ V → ( I ↾ 𝐸) ∈ V)
53, 4mp1i 13 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ( I ↾ 𝐸) ∈ V)
6 fusgrmaxsize.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
7 usgrsscusgra.h . . . . . . . 8 𝑉 = (Vtx‘𝐻)
8 usgrsscusgra.f . . . . . . . 8 𝐹 = (Edg‘𝐻)
96, 1, 7, 8sizusglecusglem1 26413 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ( I ↾ 𝐸):𝐸1-1𝐹)
10 f1eq1 6134 . . . . . . . 8 (𝑓 = ( I ↾ 𝐸) → (𝑓:𝐸1-1𝐹 ↔ ( I ↾ 𝐸):𝐸1-1𝐹))
1110spcegv 3325 . . . . . . 7 (( I ↾ 𝐸) ∈ V → (( I ↾ 𝐸):𝐸1-1𝐹 → ∃𝑓 𝑓:𝐸1-1𝐹))
125, 9, 11sylc 65 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ∃𝑓 𝑓:𝐸1-1𝐹)
1312adantl 481 . . . . 5 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → ∃𝑓 𝑓:𝐸1-1𝐹)
14 hashdom 13206 . . . . . . 7 ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → ((#‘𝐸) ≤ (#‘𝐹) ↔ 𝐸𝐹))
1514adantr 480 . . . . . 6 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → ((#‘𝐸) ≤ (#‘𝐹) ↔ 𝐸𝐹))
16 brdomg 8007 . . . . . . . 8 (𝐹 ∈ Fin → (𝐸𝐹 ↔ ∃𝑓 𝑓:𝐸1-1𝐹))
1716adantl 481 . . . . . . 7 ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → (𝐸𝐹 ↔ ∃𝑓 𝑓:𝐸1-1𝐹))
1817adantr 480 . . . . . 6 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → (𝐸𝐹 ↔ ∃𝑓 𝑓:𝐸1-1𝐹))
1915, 18bitrd 268 . . . . 5 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → ((#‘𝐸) ≤ (#‘𝐹) ↔ ∃𝑓 𝑓:𝐸1-1𝐹))
2013, 19mpbird 247 . . . 4 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → (#‘𝐸) ≤ (#‘𝐹))
2120exp31 629 . . 3 (𝐸 ∈ Fin → (𝐹 ∈ Fin → ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘𝐹))))
226, 1, 7, 8sizusglecusglem2 26414 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ∧ 𝐹 ∈ Fin) → 𝐸 ∈ Fin)
2322pm2.24d 147 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ∧ 𝐹 ∈ Fin) → (¬ 𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹)))
24233expia 1286 . . . 4 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (𝐹 ∈ Fin → (¬ 𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹))))
2524com13 88 . . 3 𝐸 ∈ Fin → (𝐹 ∈ Fin → ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘𝐹))))
2621, 25pm2.61i 176 . 2 (𝐹 ∈ Fin → ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘𝐹)))
27 fvex 6239 . . . . 5 (Edg‘𝐻) ∈ V
288, 27eqeltri 2726 . . . 4 𝐹 ∈ V
29 nfile 13188 . . . 4 ((𝐸 ∈ V ∧ 𝐹 ∈ V ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐸) ≤ (#‘𝐹))
303, 28, 29mp3an12 1454 . . 3 𝐹 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹))
3130a1d 25 . 2 𝐹 ∈ Fin → ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘𝐹)))
3226, 31pm2.61i 176 1 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  Vcvv 3231   class class class wbr 4685   I cid 5052   ↾ cres 5145  –1-1→wf1 5923  ‘cfv 5926   ≼ cdom 7995  Fincfn 7997   ≤ cle 10113  #chash 13157  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089  ComplUSGraphccusgr 26361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-umgr 26023  df-uspgr 26090  df-usgr 26091  df-fusgr 26254  df-nbgr 26270  df-uvtx 26332  df-cplgr 26362  df-cusgr 26363 This theorem is referenced by:  fusgrmaxsize  26416
 Copyright terms: Public domain W3C validator