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Theorem brafn 31669
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 ∈ β„‹ β†’ (braβ€˜π΄): β„‹βŸΆβ„‚)

Proof of Theorem brafn
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 brafval 31665 . 2 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
2 hicl 30802 . . 3 ((π‘₯ ∈ β„‹ ∧ 𝐴 ∈ β„‹) β†’ (π‘₯ Β·ih 𝐴) ∈ β„‚)
32ancoms 458 . 2 ((𝐴 ∈ β„‹ ∧ π‘₯ ∈ β„‹) β†’ (π‘₯ Β·ih 𝐴) ∈ β„‚)
41, 3fmpt3d 7107 1 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄): β„‹βŸΆβ„‚)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401  β„‚cc 11104   β„‹chba 30641   Β·ih csp 30644  bracbr 30678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-hilex 30721  ax-hfi 30801
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-bra 31572
This theorem is referenced by:  bralnfn  31670  bracl  31671  brafnmul  31673  branmfn  31827  rnbra  31829  kbass2  31839  kbass3  31840
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