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Theorem brafn 31200
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 31196 . 2 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
2 hicl 30333 . . 3 ((π‘₯ ∈ β„‹ ∧ 𝐴 ∈ β„‹) β†’ (π‘₯ Β·ih 𝐴) ∈ β„‚)
32ancoms 460 . 2 ((𝐴 ∈ β„‹ ∧ π‘₯ ∈ β„‹) β†’ (π‘₯ Β·ih 𝐴) ∈ β„‚)
41, 3fmpt3d 7116 1 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄): β„‹βŸΆβ„‚)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108   β„‹chba 30172   Β·ih csp 30175  bracbr 30209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-hilex 30252  ax-hfi 30332
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-bra 31103
This theorem is referenced by:  bralnfn  31201  bracl  31202  brafnmul  31204  branmfn  31358  rnbra  31360  kbass2  31370  kbass3  31371
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