Theorem opiedgfv 26795

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 5598	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opiedgval 26794	. . 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 7705	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
5	3, 4	eqtrd 2859	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3497  ⟨cop 4576   × cxp 5556  ‘cfv 6358  2^nd c2nd 7691  iEdgciedg 26785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-iota 6317  df-fun 6360  df-fv 6366  df-2nd 7693  df-iedg 26787

This theorem is referenced by:  opiedgov  26796  opiedgfvi  26798  gropd  26819  edgopval  26839  isuhgrop  26858  uhgrunop  26863  upgrop  26882  upgr0eop  26902  upgr1eop  26903  upgrunop  26907  umgrunop  26909  isuspgrop  26949  isusgrop  26950  ausgrusgrb  26953  usgr0eop  27031  uspgr1eop  27032  usgr1eop  27035  usgrexmpllem  27045  uhgrspan1lem3  27087  upgrres1lem3  27097  fusgrfisbase  27113  fusgrfisstep  27114  usgrexi  27226  cusgrexi  27228  p1evtxdeqlem  27297  p1evtxdeq  27298  p1evtxdp1  27299  uspgrloopiedg  27302  umgr2v2eiedg  27308  wlk2v2e  27939  eupthvdres  28017  eupth2lemb  28019  konigsbergiedg  28029  strisomgrop  44012  ushrisomgr  44013