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Theorem brafn 29146
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 28277 . . . 4 ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ)
21ancoms 455 . . 3 ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ)
3 eqid 2771 . . 3 (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))
42, 3fmptd 6527 . 2 (𝐴 ∈ ℋ → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)): ℋ⟶ℂ)
5 brafval 29142 . . 3 (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
65feq1d 6170 . 2 (𝐴 ∈ ℋ → ((bra‘𝐴): ℋ⟶ℂ ↔ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)): ℋ⟶ℂ))
74, 6mpbird 247 1 (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  cc 10136  chil 28116   ·ih csp 28119  bracbr 28153
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-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-hilex 28196  ax-hfi 28276
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-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  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-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-bra 29049
This theorem is referenced by:  bralnfn  29147  bracl  29148  brafnmul  29150  branmfn  29304  rnbra  29306  kbass2  29316  kbass3  29317
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