Theorem opiedgfvi 28538

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, 4-Mar-2021.)

Hypotheses

Ref	Expression
opvtxfvi.v	⊢ 𝑉 ∈ V
opvtxfvi.e	⊢ 𝐸 ∈ V

Assertion

Ref	Expression
opiedgfvi	⊢ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸

Proof of Theorem opiedgfvi

Step	Hyp	Ref	Expression
1		opvtxfvi.v	. 2 ⊢ 𝑉 ∈ V
2		opvtxfvi.e	. 2 ⊢ 𝐸 ∈ V
3		opiedgfv 28535	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
4	1, 2, 3	mp2an 689	1 ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸

Syntax hints:   = wceq 1540   ∈ wcel 2105  Vcvv 3473  ⟨cop 4634  ‘cfv 6543  iEdgciedg 28525

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-2nd 7979  df-iedg 28527

This theorem is referenced by:  graop  28557  iedgvalsnop  28570  griedg0ssusgr  28790  uhgrspanop  28821  vtxdgop  28995  vtxdginducedm1lem1  29064  finsumvtxdg2size  29075  rgrusgrprc  29114  eupth2lem3  29757  konigsberglem1  29773  konigsberglem2  29774  konigsberglem3  29775