Theorem opiedgfvi 28705

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 28702	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
4	1, 2, 3	mp2an 689	1 ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3466  ⟨cop 4626  ‘cfv 6533  iEdgciedg 28692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fv 6541  df-2nd 7969  df-iedg 28694

This theorem is referenced by:  graop  28724  iedgvalsnop  28737  griedg0ssusgr  28957  uhgrspanop  28988  vtxdgop  29162  vtxdginducedm1lem1  29231  finsumvtxdg2size  29242  rgrusgrprc  29281  eupth2lem3  29924  konigsberglem1  29940  konigsberglem2  29941  konigsberglem3  29942