Theorem brafn 28806

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.

Step	Hyp	Ref	Expression
1		hicl 27937	. . . 4 |- ( ( x e. ~H /\ A e. ~H ) -> ( x .ih A ) e. CC )
2	1	ancoms 469	. . 3 |- ( ( A e. ~H /\ x e. ~H ) -> ( x .ih A ) e. CC )
3		eqid 2622	. . 3 |- ( x e. ~H |-> ( x .ih A ) ) = ( x e. ~H |-> ( x .ih A ) )
4	2, 3	fmptd 6385	. 2 |- ( A e. ~H -> ( x e. ~H |-> ( x .ih A ) ) : ~H --> CC )
5		brafval 28802	. . 3 |- ( A e. ~H -> ( bra ` A ) = ( x e. ~H |-> ( x .ih A ) ) )
6	5	feq1d 6030	. 2 |- ( A e. ~H -> ( ( bra ` A ) : ~H --> CC <-> ( x e. ~H |-> ( x .ih A ) ) : ~H --> CC ) )
7	4, 6	mpbird 247	1 |- ( A e. ~H -> ( bra ` A ) : ~H --> CC )

Syntax hints:   -> wi 4   e. wcel 1990   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   ~Hchil 27776   .ih csp 27779   bracbr 27813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hilex 27856  ax-hfi 27936

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-bra 28709

This theorem is referenced by:  bralnfn  28807  bracl  28808  brafnmul  28810  branmfn  28964  rnbra  28966  kbass2  28976  kbass3  28977