MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  upgr0eopALT Structured version   Visualization version   GIF version

Theorem upgr0eopALT 29134
Description: Alternate proof of upgr0eop 29132, using the general theorem gropeld 29051 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr0eop 29132). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
upgr0eopALT (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Proof of Theorem upgr0eopALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 vex 3483 . . . . . 6 𝑔 ∈ V
21a1i 11 . . . . 5 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V)
3 simpr 484 . . . . 5 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅)
42, 3upgr0e 29129 . . . 4 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
54ax-gen 1794 . . 3 𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
65a1i 11 . 2 (𝑉𝑊 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph))
7 id 22 . 2 (𝑉𝑊𝑉𝑊)
8 0ex 5306 . . 3 ∅ ∈ V
98a1i 11 . 2 (𝑉𝑊 → ∅ ∈ V)
106, 7, 9gropeld 29051 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1537   = wceq 1539  wcel 2107  Vcvv 3479  c0 4332  cop 4631  cfv 6560  Vtxcvtx 29014  iEdgciedg 29015  UPGraphcupgr 29098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-i2m1 11224  ax-1ne0 11225  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-1st 8015  df-2nd 8016  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-2 12330  df-vtx 29016  df-iedg 29017  df-upgr 29100  df-umgr 29101
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator