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Theorem opvtxfv 26476
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 5490 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opvtxval 26475 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
4 op1stg 7564 . 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 396   = wceq 1525  wcel 2083  Vcvv 3440  cop 4484   × cxp 5448  cfv 6232  1st c1st 7550  Vtxcvtx 26468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-iota 6196  df-fun 6234  df-fv 6240  df-1st 7552  df-vtx 26470
This theorem is referenced by:  opvtxov  26477  opvtxfvi  26481  gropd  26503  isuhgrop  26542  uhgrunop  26547  upgrop  26566  upgr1eop  26587  upgrunop  26591  umgrunop  26593  isuspgrop  26633  isusgrop  26634  ausgrusgrb  26637  uspgr1eop  26716  usgr1eop  26719  usgrexmpllem  26729  uhgrspan1lem2  26770  upgrres1lem2  26780  opfusgr  26792  fusgrfisbase  26797  fusgrfisstep  26798  usgrexi  26910  cusgrexi  26912  p1evtxdeqlem  26981  p1evtxdeq  26982  p1evtxdp1  26983  uspgrloopvtx  26984  umgr2v2evtx  26990  wlk2v2e  27622  eupthvdres  27700  eupth2lemb  27702  konigsbergvtx  27711  konigsberg  27722  strisomgrop  43509  ushrisomgr  43510  uspgrsprfo  43527
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