Theorem opiedgfv 29450

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 5700	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opiedgval 29449	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ (V × V) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
3	1, 2	syl 18	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
4		op2ndg 8002	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
5	3, 4	eqtrd 2797	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593   × cxp 5657  ‘cfv 6537  2^nd c2nd 7988  iEdgciedg 29440

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fv 6545  df-2nd 7990  df-iedg 29442

This theorem is used by:  opiedgov  29451  opiedgfvi  29453  gropd  29474  edgopval  29494  isuhgrop  29513  uhgrunop  29518  upgrop  29537  upgr0eop  29557  upgr1eop  29558  upgrunop  29562  umgrunop  29564  isuspgrop  29607  isusgrop  29608  ausgrusgrb  29611  usgr0eop  29692  uspgr1eop  29693  usgr1eop  29696  usgrexmpllem  29706  uhgrspan1lem3  29748  upgrres1lem3  29758  fusgrfisbase  29774  fusgrfisstep  29775  usgrexi  29887  cusgrexi  29889  p1evtxdeqlem  29958  p1evtxdeq  29959  p1evtxdp1  29960  uspgrloopiedg  29963  umgr2v2eiedg  29969  wlk2v2e  30623  eupthvdres  30701  eupth2lemb  30703  konigsbergiedg  30713  isubgriedg  48766  opstrgric  48829  ushggricedg  48830  usgrexmpl1edg  48927  usgrexmpl2edg  48932