Theorem opiedgfvi 29618

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 29615	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
4	1, 2, 3	mp2an 705	1 ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838  ‘cfv 6558  iEdgciedg 29605

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-2nd 8026  df-iedg 29607

This theorem is used by:  graop  29637  iedgvalsnop  29650  griedg0ssusgr  29876  uhgrspanop  29907  vtxdgop  30081  vtxdginducedm1lem1  30150  finsumvtxdg2size  30161  rgrusgrprc  30200  eupth2lem3  30867  konigsberglem1  30883  konigsberglem2  30884  konigsberglem3  30885