Theorem opiedgfvi 28991

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 28988	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
4	1, 2, 3	mp2an 692	1 ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591  ‘cfv 6499  iEdgciedg 28978

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fv 6507  df-2nd 7948  df-iedg 28980

This theorem is referenced by:  graop  29010  iedgvalsnop  29023  griedg0ssusgr  29246  uhgrspanop  29277  vtxdgop  29452  vtxdginducedm1lem1  29521  finsumvtxdg2size  29532  rgrusgrprc  29571  eupth2lem3  30216  konigsberglem1  30232  konigsberglem2  30233  konigsberglem3  30234