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Theorem vcex 27773
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 4787 . 2 (𝐺CVecOLD𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ CVecOLD)
2 vcrel 27755 . . . 4 Rel CVecOLD
3 df-rel 5256 . . . 4 (Rel CVecOLD ↔ CVecOLD ⊆ (V × V))
42, 3mpbi 220 . . 3 CVecOLD ⊆ (V × V)
54brel 5308 . 2 (𝐺CVecOLD𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
61, 5sylbir 225 1 (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  Vcvv 3351  wss 3723  cop 4322   class class class wbr 4786   × cxp 5247  Rel wrel 5254  CVecOLDcvc 27753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-vc 27754
This theorem is referenced by:  isvcOLD  27774  nvex  27806  isnv  27807
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