Theorem opiedgfv 28992

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	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Proof of Theorem opiedgfv

Step	Hyp	Ref	Expression
1		opelvvg 5660	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opiedgval 28991	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ (V × V) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
3	1, 2	syl 17	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
4		op2ndg 7940	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
5	3, 4	eqtrd 2766	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4581   × cxp 5617  ‘cfv 6487  2^nd c2nd 7926  iEdgciedg 28982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6443  df-fun 6489  df-fv 6495  df-2nd 7928  df-iedg 28984

This theorem is referenced by:  opiedgov  28993  opiedgfvi  28995  gropd  29016  edgopval  29036  isuhgrop  29055  uhgrunop  29060  upgrop  29079  upgr0eop  29099  upgr1eop  29100  upgrunop  29104  umgrunop  29106  isuspgrop  29146  isusgrop  29147  ausgrusgrb  29150  usgr0eop  29231  uspgr1eop  29232  usgr1eop  29235  usgrexmpllem  29245  uhgrspan1lem3  29287  upgrres1lem3  29297  fusgrfisbase  29313  fusgrfisstep  29314  usgrexi  29426  cusgrexi  29428  p1evtxdeqlem  29498  p1evtxdeq  29499  p1evtxdp1  29500  uspgrloopiedg  29503  umgr2v2eiedg  29509  wlk2v2e  30144  eupthvdres  30222  eupth2lemb  30224  konigsbergiedg  30234  isubgriedg  47968  opstrgric  48031  ushggricedg  48032  usgrexmpl1edg  48129  usgrexmpl2edg  48134