Theorem opiedgfv 26305

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 5382	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opiedgval 26304	. . 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 7441	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
5	3, 4	eqtrd 2861	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  Vcvv 3414  ⟨cop 4403   × cxp 5340  ‘cfv 6123  2^nd c2nd 7427  iEdgciedg 26295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fv 6131  df-2nd 7429  df-iedg 26297

This theorem is referenced by:  opiedgov  26306  opiedgfvi  26308  graop  26327  gropd  26329  edgopval  26349  isuhgrop  26368  uhgrunop  26373  upgrop  26392  upgr0eop  26412  upgr1eop  26413  upgrunop  26417  umgrunop  26419  isuspgrop  26460  isusgrop  26461  ausgrusgrb  26464  usgr0eop  26543  uspgr1eop  26544  usgr1eop  26547  usgrexmpllem  26557  griedg0ssusgr  26562  uhgrspanop  26593  uhgrspan1lem3  26599  upgrres1lem3  26609  fusgrfisbase  26625  fusgrfisstep  26626  usgrexi  26739  cusgrexi  26741  p1evtxdeqlem  26810  p1evtxdeq  26811  p1evtxdp1  26812  uspgrloopiedg  26815  umgr2v2eiedg  26821  rgrusgrprc  26887  wlk2v2e  27522  eupthvdres  27601  eupth2lemb  27603  konigsbergiedg  27615  strisomgrop  42551  ushrisomgr  42552