Theorem opiedgfv 29085

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 5666	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opiedgval 29084	. . 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 7949	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
5	3, 4	eqtrd 2772	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441  ⟨cop 4587   × cxp 5623  ‘cfv 6493  2^nd c2nd 7935  iEdgciedg 29075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fv 6501  df-2nd 7937  df-iedg 29077

This theorem is referenced by:  opiedgov  29086  opiedgfvi  29088  gropd  29109  edgopval  29129  isuhgrop  29148  uhgrunop  29153  upgrop  29172  upgr0eop  29192  upgr1eop  29193  upgrunop  29197  umgrunop  29199  isuspgrop  29239  isusgrop  29240  ausgrusgrb  29243  usgr0eop  29324  uspgr1eop  29325  usgr1eop  29328  usgrexmpllem  29338  uhgrspan1lem3  29380  upgrres1lem3  29390  fusgrfisbase  29406  fusgrfisstep  29407  usgrexi  29519  cusgrexi  29521  p1evtxdeqlem  29591  p1evtxdeq  29592  p1evtxdp1  29593  uspgrloopiedg  29596  umgr2v2eiedg  29602  wlk2v2e  30237  eupthvdres  30315  eupth2lemb  30317  konigsbergiedg  30327  isubgriedg  48186  opstrgric  48249  ushggricedg  48250  usgrexmpl1edg  48347  usgrexmpl2edg  48352