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Theorem vcex 26817
Description: The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Assertion
Ref Expression
vcex (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))

Proof of Theorem vcex
StepHypRef Expression
1 df-br 4584 . 2 (𝐺CVecOLD𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ CVecOLD)
2 vcrel 26799 . . . 4 Rel CVecOLD
3 df-rel 5045 . . . 4 (Rel CVecOLD ↔ CVecOLD ⊆ (V × V))
42, 3mpbi 219 . . 3 CVecOLD ⊆ (V × V)
54brel 5090 . 2 (𝐺CVecOLD𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
61, 5sylbir 224 1 (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  Vcvv 3173  wss 3540  cop 4131   class class class wbr 4583   × cxp 5036  Rel wrel 5043  CVecOLDcvc 26797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-vc 26798
This theorem is referenced by:  isvcOLD  26818  nvex  26850  isnv  26851
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