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

Theorem opvtxfv 26239
Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
Assertion
Ref Expression
opvtxfv ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Proof of Theorem opvtxfv
StepHypRef Expression
1 opelvvg 5352 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opvtxval 26238 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
4 op1stg 7413 . 2 ((𝑉𝑋𝐸𝑌) → (1st ‘⟨𝑉, 𝐸⟩) = 𝑉)
53, 4eqtrd 2833 1 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  cop 4374   × cxp 5310  cfv 6101  1st c1st 7399  Vtxcvtx 26231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fv 6109  df-1st 7401  df-vtx 26233
This theorem is referenced by:  opvtxov  26240  opvtxfvi  26244  graop  26264  gropd  26266  isuhgrop  26305  uhgrunop  26310  upgrop  26329  upgr1eop  26350  upgrunop  26354  umgrunop  26356  isuspgrop  26397  isusgrop  26398  ausgrusgrb  26401  uspgr1eop  26481  usgr1eop  26484  usgrexmpllem  26494  uhgrspanop  26530  uhgrspan1lem2  26535  upgrres1lem2  26545  opfusgr  26557  fusgrfisbase  26562  fusgrfisstep  26563  usgrexi  26691  cusgrexi  26693  fusgrmaxsize  26714  p1evtxdeqlem  26762  p1evtxdeq  26763  p1evtxdp1  26764  uspgrloopvtx  26765  umgr2v2evtx  26771  wlk2v2e  27501  eupthvdres  27580  eupth2lemb  27582  konigsbergvtx  27593  konigsberg  27604  strisomgrop  42510  ushrisomgr  42511  uspgrsprfo  42555
  Copyright terms: Public domain W3C validator