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Theorem brafn 28190
 Description: The bra function is a functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
brafn (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)

Proof of Theorem brafn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hicl 27321 . . . 4 ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ)
21ancoms 468 . . 3 ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ)
3 eqid 2610 . . 3 (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))
42, 3fmptd 6292 . 2 (𝐴 ∈ ℋ → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)): ℋ⟶ℂ)
5 brafval 28186 . . 3 (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
65feq1d 5943 . 2 (𝐴 ∈ ℋ → ((bra‘𝐴): ℋ⟶ℂ ↔ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)): ℋ⟶ℂ))
74, 6mpbird 246 1 (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   ℋchil 27160   ·ih csp 27163  bracbr 27197 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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-hilex 27240  ax-hfi 27320 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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-bra 28093 This theorem is referenced by:  bralnfn  28191  bracl  28192  brafnmul  28194  branmfn  28348  rnbra  28350  kbass2  28360  kbass3  28361
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