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Theorem opiedgfv 26007
Description: The set of indexed edges 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
opiedgfv ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Proof of Theorem opiedgfv
StepHypRef Expression
1 opelvvg 5274 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opiedgval 26006 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
4 op2ndg 7298 . 2 ((𝑉𝑋𝐸𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
53, 4eqtrd 2758 1 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  Vcvv 3304  cop 4291   × cxp 5216  cfv 6001  2nd c2nd 7284  iEdgciedg 25995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fv 6009  df-2nd 7286  df-iedg 25997
This theorem is referenced by:  opiedgov  26008  opiedgfvi  26010  graop  26041  gropd  26043  edgopval  26064  isuhgrop  26085  uhgrunop  26090  upgrop  26109  upgr0eop  26129  upgr1eop  26130  upgrunop  26134  umgrunop  26136  isuspgrop  26176  isusgrop  26177  ausgrusgrb  26180  usgr0eop  26258  uspgr1eop  26259  usgr1eop  26262  usgrexmpllem  26272  griedg0ssusgr  26277  uhgrspanop  26308  uhgrspan1lem3  26314  upgrres1lem3  26324  fusgrfisbase  26340  fusgrfisstep  26341  usgrexi  26468  cusgrexi  26470  p1evtxdeqlem  26539  p1evtxdeq  26540  p1evtxdp1  26541  uspgrloopiedg  26544  umgr2v2eiedg  26550  rgrusgrprc  26616  wlk2v2e  27230  eupthvdres  27308  eupth2lemb  27310  konigsbergiedg  27320
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