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Theorem bj-charfundc 13425
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 6390 . . . . 5  |-  1o  e.  2o
32a1i 9 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  1o  e.  2o )
4 0lt2o 6389 . . . . 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 2545 . . . 4  |-  ( (
ph  /\  x  e.  X )  -> DECID  x  e.  A
)
83, 5, 7ifcldcd 3540 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  if ( x  e.  A ,  1o ,  (/) )  e.  2o )
91, 8fmpt3d 5624 . 2  |-  ( ph  ->  F : X --> 2o )
10 inss1 3327 . . . . . . . 8  |-  ( X  i^i  A )  C_  X
1110a1i 9 . . . . . . 7  |-  ( ph  ->  ( X  i^i  A
)  C_  X )
1211sseld 3127 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  i^i  A )  ->  x  e.  X
) )
1312imdistani 442 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  ( ph  /\  x  e.  X ) )
141, 8fvmpt2d 5555 . . . . 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 3297 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  x  e.  A )
1817iftrued 3512 . . . 4  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  if (
x  e.  A ,  1o ,  (/) )  =  1o )
1915, 18eqtrd 2190 . . 3  |-  ( (
ph  /\  x  e.  ( X  i^i  A ) )  ->  ( F `  x )  =  1o )
2019ralrimiva 2530 . 2  |-  ( ph  ->  A. x  e.  ( X  i^i  A ) ( F `  x
)  =  1o )
21 difssd 3234 . . . . . . 7  |-  ( ph  ->  ( X  \  A
)  C_  X )
2221sseld 3127 . . . . . 6  |-  ( ph  ->  ( x  e.  ( X  \  A )  ->  x  e.  X
) )
2322imdistani 442 . . . . 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 3114 . . . . 5  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  -.  x  e.  A )
2726iffalsed 3515 . . . 4  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  if (
x  e.  A ,  1o ,  (/) )  =  (/) )
2824, 27eqtrd 2190 . . 3  |-  ( (
ph  /\  x  e.  ( X  \  A ) )  ->  ( F `  x )  =  (/) )
2928ralrimiva 2530 . 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 820    = wceq 1335    e. wcel 2128   A.wral 2435    \ cdif 3099    i^i cin 3101    C_ wss 3102   (/)c0 3394   ifcif 3505    |-> cmpt 4026   -->wf 5167   ` cfv 5171   1oc1o 6357   2oc2o 6358
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-suc 4332  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-fv 5179  df-1o 6364  df-2o 6365
This theorem is referenced by:  bj-charfunbi  13428
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