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Theorem ddeval1 28402
Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
Assertion
Ref Expression
ddeval1  |-  ( ( A  C_  RR  /\  0  e.  A )  ->  (δ `  A )  =  1 )

Proof of Theorem ddeval1
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 reex 9516 . . . . 5  |-  RR  e.  _V
21ssex 4526 . . . 4  |-  ( A 
C_  RR  ->  A  e. 
_V )
3 elpwg 3952 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P RR  <->  A 
C_  RR ) )
43biimpar 483 . . . 4  |-  ( ( A  e.  _V  /\  A  C_  RR )  ->  A  e.  ~P RR )
52, 4mpancom 667 . . 3  |-  ( A 
C_  RR  ->  A  e. 
~P RR )
6 eleq2 2469 . . . . 5  |-  ( a  =  A  ->  (
0  e.  a  <->  0  e.  A ) )
76ifbid 3896 . . . 4  |-  ( a  =  A  ->  if ( 0  e.  a ,  1 ,  0 )  =  if ( 0  e.  A , 
1 ,  0 ) )
8 df-dde 28401 . . . 4  |- δ  =  ( a  e.  ~P RR  |->  if ( 0  e.  a ,  1 ,  0 ) )
9 1ex 9524 . . . . 5  |-  1  e.  _V
10 c0ex 9523 . . . . 5  |-  0  e.  _V
119, 10ifex 3942 . . . 4  |-  if ( 0  e.  A , 
1 ,  0 )  e.  _V
127, 8, 11fvmpt 5874 . . 3  |-  ( A  e.  ~P RR  ->  (δ `  A )  =  if ( 0  e.  A ,  1 ,  0 ) )
135, 12syl 16 . 2  |-  ( A 
C_  RR  ->  (δ `  A
)  =  if ( 0  e.  A , 
1 ,  0 ) )
14 iftrue 3880 . 2  |-  ( 0  e.  A  ->  if ( 0  e.  A ,  1 ,  0 )  =  1 )
1513, 14sylan9eq 2457 1  |-  ( ( A  C_  RR  /\  0  e.  A )  ->  (δ `  A )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   _Vcvv 3051    C_ wss 3406   ifcif 3874   ~Pcpw 3944   ` cfv 5513   RRcr 9424   0cc0 9425   1c1 9426  δcdde 28400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-mulcl 9487  ax-i2m1 9493
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-iota 5477  df-fun 5515  df-fv 5521  df-dde 28401
This theorem is referenced by:  ddemeas  28404
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