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Theorem brafn 22543
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  e.  ~H  ->  ( bra `  A ) : ~H --> CC )

Proof of Theorem brafn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hicl 21675 . . . 4  |-  ( ( x  e.  ~H  /\  A  e.  ~H )  ->  ( x  .ih  A
)  e.  CC )
21ancoms 439 . . 3  |-  ( ( A  e.  ~H  /\  x  e.  ~H )  ->  ( x  .ih  A
)  e.  CC )
3 eqid 2296 . . 3  |-  ( x  e.  ~H  |->  ( x 
.ih  A ) )  =  ( x  e. 
~H  |->  ( x  .ih  A ) )
42, 3fmptd 5700 . 2  |-  ( A  e.  ~H  ->  (
x  e.  ~H  |->  ( x  .ih  A ) ) : ~H --> CC )
5 brafval 22539 . . 3  |-  ( A  e.  ~H  ->  ( bra `  A )  =  ( x  e.  ~H  |->  ( x  .ih  A ) ) )
65feq1d 5395 . 2  |-  ( A  e.  ~H  ->  (
( bra `  A
) : ~H --> CC  <->  ( x  e.  ~H  |->  ( x  .ih  A ) ) : ~H --> CC ) )
74, 6mpbird 223 1  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   ~Hchil 21515    .ih csp 21518   bracbr 21552
This theorem is referenced by:  bralnfn  22544  bracl  22545  brafnmul  22547  branmfn  22701  rnbra  22703  kbass2  22713  kbass3  22714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595  ax-hfi 21674
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-bra 22446
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