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Theorem brafn 29740
 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 brafval 29736 . 2 (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
2 hicl 28873 . . 3 ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ)
32ancoms 462 . 2 ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ)
41, 3fmpt3d 6858 1 (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  ℂcc 10527   ℋchba 28712   ·ih csp 28715  bracbr 28749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-hilex 28792  ax-hfi 28872 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-bra 29643 This theorem is referenced by:  bralnfn  29741  bracl  29742  brafnmul  29744  branmfn  29898  rnbra  29900  kbass2  29910  kbass3  29911
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