Theorem brafn 31966

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.

Step	Hyp	Ref	Expression
1		brafval 31962	. 2 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
2		hicl 31099	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ)
3	2	ancoms 458	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ)
4	1, 3	fmpt3d 7136	1 ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)

Syntax hints:   → wi 4   ∈ wcel 2108  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℂcc 11153   ℋchba 30938   ·_ih csp 30941  bracbr 30975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018  ax-hfi 31098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-bra 31869

This theorem is referenced by:  bralnfn  31967  bracl  31968  brafnmul  31970  branmfn  32124  rnbra  32126  kbass2  32136  kbass3  32137