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Theorem vcrel 30589
Description: The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Assertion
Ref Expression
vcrel Rel CVecOLD

Proof of Theorem vcrel
Dummy variables 𝑔 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vc 30588 . 2 CVecOLD = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))}
21relopabiv 5833 1 Rel CVecOLD
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059   × cxp 5687  ran crn 5690  Rel wrel 5694  wf 6559  (class class class)co 7431  cc 11151  1c1 11154   + caddc 11156   · cmul 11158  AbelOpcablo 30573  CVecOLDcvc 30587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695  df-rel 5696  df-vc 30588
This theorem is referenced by:  vcex  30607  nvvop  30638  phop  30847
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