Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-charfundc Unicode version

Theorem bj-charfundc 13843
Description: Properties of the characteristic function on the class  X of the class  A, provided membership in  A is decidable in  X. (Contributed by BJ, 6-Aug-2024.)
Hypotheses
Ref Expression
bj-charfundc.1  |-  ( ph  ->  F  =  ( x  e.  X  |->  if ( x  e.  A ,  1o ,  (/) ) ) )
bj-charfundc.dc  |-  ( ph  ->  A. x  e.  X DECID  x  e.  A )
Assertion
Ref Expression
bj-charfundc  |-  ( ph  ->  ( F : X --> 2o  /\  ( A. x  e.  ( X  i^i  A
) ( F `  x )  =  1o 
/\  A. x  e.  ( X  \  A ) ( F `  x
)  =  (/) ) ) )
Distinct variable groups:    ph, x    x, X
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem bj-charfundc
StepHypRef Expression
1 bj-charfundc.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  X  |->  if ( x  e.  A ,  1o ,  (/) ) ) )
2 1lt2o 6421 . . . . 5  |-  1o  e.  2o
32a1i 9 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  1o  e.  2o )
4 0lt2o 6420 . . . . 5  |-  (/)  e.  2o
54a1i 9 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (/)  e.  2o )
6 bj-charfundc.dc . . . . 5  |-  ( ph  ->  A. x  e.  X DECID  x  e.  A )
76r19.21bi 2558 . . . 4  |-  ( (
ph  /\  x  e.  X )  -> DECID  x  e.  A
)
83, 5, 7ifcldcd 3561 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  if ( x  e.  A ,  1o ,  (/) )  e.  2o )
91, 8fmpt3d 5652 . 2  |-  ( ph  ->  F : X --> 2o )
10 inss1 3347 . . . . . . . 8  |-  ( X  i^i  A )  C_  X
1110a1i 9 . . . . . . 7  |-  ( ph  ->  ( X  i^i  A
)  C_  X )
1211sseld 3146 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  i^i  A )  ->  x  e.  X
) )
1312imdistani 443 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  ( ph  /\  x  e.  X ) )
141, 8fvmpt2d 5582 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  =  if ( x  e.  A ,  1o ,  (/) ) )
1513, 14syl 14 . . . 4  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  ( F `  x )  =  if ( x  e.  A ,  1o ,  (/) ) )
16 simpr 109 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  x  e.  ( X  i^i  A ) )
1716elin2d 3317 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  x  e.  A )
1817iftrued 3533 . . . 4  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  if (
x  e.  A ,  1o ,  (/) )  =  1o )
1915, 18eqtrd 2203 . . 3  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  ( F `  x )  =  1o )
2019ralrimiva 2543 . 2  |-  ( ph  ->  A. x  e.  ( X  i^i  A ) ( F `  x
)  =  1o )
21 difssd 3254 . . . . . . 7  |-  ( ph  ->  ( X  \  A
)  C_  X )
2221sseld 3146 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  \  A )  ->  x  e.  X
) )
2322imdistani 443 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  ( ph  /\  x  e.  X ) )
2423, 14syl 14 . . . 4  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  ( F `  x )  =  if ( x  e.  A ,  1o ,  (/) ) )
25 simpr 109 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  x  e.  ( X  \  A ) )
2625eldifbd 3133 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  -.  x  e.  A )
2726iffalsed 3536 . . . 4  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  if (
x  e.  A ,  1o ,  (/) )  =  (/) )
2824, 27eqtrd 2203 . . 3  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  ( F `  x )  =  (/) )
2928ralrimiva 2543 . 2  |-  ( ph  ->  A. x  e.  ( X  \  A ) ( F `  x
)  =  (/) )
309, 20, 29jca32 308 1  |-  ( ph  ->  ( F : X --> 2o  /\  ( A. x  e.  ( X  i^i  A
) ( F `  x )  =  1o 
/\  A. x  e.  ( X  \  A ) ( F `  x
)  =  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103  DECID wdc 829    = wceq 1348    e. wcel 2141   A.wral 2448    \ cdif 3118    i^i cin 3120    C_ wss 3121   (/)c0 3414   ifcif 3526    |-> cmpt 4050   -->wf 5194   ` cfv 5198   1oc1o 6388   2oc2o 6389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-1o 6395  df-2o 6396
This theorem is referenced by:  bj-charfunbi  13846
  Copyright terms: Public domain W3C validator