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Theorem bj-charfundcALT 15371
Description: Alternate proof of bj-charfundc 15370. It was expected to be much shorter since it uses bj-charfun 15369 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
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-charfundcALT  |-  ( 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    x, A    x, F

Proof of Theorem bj-charfundcALT
StepHypRef Expression
1 bj-charfundc.1 . . 3  |-  ( ph  ->  F  =  ( x  e.  X  |->  if ( x  e.  A ,  1o ,  (/) ) ) )
21bj-charfun 15369 . 2  |-  ( ph  ->  ( ( F : X
--> ~P 1o  /\  ( F  |`  ( ( X  i^i  A )  u.  ( X  \  A
) ) ) : ( ( X  i^i  A )  u.  ( X 
\  A ) ) --> 2o )  /\  ( A. x  e.  ( X  i^i  A ) ( F `  x )  =  1o  /\  A. x  e.  ( X  \  A ) ( F `
 x )  =  (/) ) ) )
3 difin 3397 . . . . . . . . . . . 12  |-  ( X 
\  ( X  i^i  A ) )  =  ( X  \  A )
43eqcomi 2197 . . . . . . . . . . 11  |-  ( X 
\  A )  =  ( X  \  ( X  i^i  A ) )
54a1i 9 . . . . . . . . . 10  |-  ( ph  ->  ( X  \  A
)  =  ( X 
\  ( X  i^i  A ) ) )
65uneq2d 3314 . . . . . . . . 9  |-  ( ph  ->  ( ( X  i^i  A )  u.  ( X 
\  A ) )  =  ( ( X  i^i  A )  u.  ( X  \  ( X  i^i  A ) ) ) )
7 inss1 3380 . . . . . . . . . . 11  |-  ( X  i^i  A )  C_  X
87a1i 9 . . . . . . . . . 10  |-  ( ph  ->  ( X  i^i  A
)  C_  X )
9 bj-charfundc.dc . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  X DECID  x  e.  A )
10 elin 3343 . . . . . . . . . . . . . 14  |-  ( x  e.  ( X  i^i  A )  <->  ( x  e.  X  /\  x  e.  A ) )
1110baibr 921 . . . . . . . . . . . . 13  |-  ( x  e.  X  ->  (
x  e.  A  <->  x  e.  ( X  i^i  A ) ) )
1211dcbid 839 . . . . . . . . . . . 12  |-  ( x  e.  X  ->  (DECID  x  e.  A  <-> DECID  x  e.  ( X  i^i  A ) ) )
1312ralbiia 2508 . . . . . . . . . . 11  |-  ( A. x  e.  X DECID  x  e.  A 
<-> 
A. x  e.  X DECID  x  e.  ( X  i^i  A
) )
149, 13sylib 122 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  X DECID  x  e.  ( X  i^i  A
) )
15 undifdcss 6981 . . . . . . . . . 10  |-  ( X  =  ( ( X  i^i  A )  u.  ( X  \  ( X  i^i  A ) ) )  <->  ( ( X  i^i  A )  C_  X  /\  A. x  e.  X DECID  x  e.  ( X  i^i  A ) ) )
168, 14, 15sylanbrc 417 . . . . . . . . 9  |-  ( ph  ->  X  =  ( ( X  i^i  A )  u.  ( X  \ 
( X  i^i  A
) ) ) )
176, 16eqtr4d 2229 . . . . . . . 8  |-  ( ph  ->  ( ( X  i^i  A )  u.  ( X 
\  A ) )  =  X )
1817reseq2d 4943 . . . . . . 7  |-  ( ph  ->  ( F  |`  (
( X  i^i  A
)  u.  ( X 
\  A ) ) )  =  ( F  |`  X ) )
19 ssidd 3201 . . . . . . . . 9  |-  ( ph  ->  X  C_  X )
2019resmptd 4994 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  X  |->  if ( x  e.  A ,  1o ,  (/) ) )  |`  X )  =  ( x  e.  X  |->  if ( x  e.  A ,  1o ,  (/) ) ) )
211reseq1d 4942 . . . . . . . 8  |-  ( ph  ->  ( F  |`  X )  =  ( ( x  e.  X  |->  if ( x  e.  A ,  1o ,  (/) ) )  |`  X ) )
2220, 21, 13eqtr4d 2236 . . . . . . 7  |-  ( ph  ->  ( F  |`  X )  =  F )
2318, 22eqtrd 2226 . . . . . 6  |-  ( ph  ->  ( F  |`  (
( X  i^i  A
)  u.  ( X 
\  A ) ) )  =  F )
2423, 17feq12d 5394 . . . . 5  |-  ( ph  ->  ( ( F  |`  ( ( X  i^i  A )  u.  ( X 
\  A ) ) ) : ( ( X  i^i  A )  u.  ( X  \  A ) ) --> 2o  <->  F : X --> 2o ) )
2524biimpd 144 . . . 4  |-  ( ph  ->  ( ( F  |`  ( ( X  i^i  A )  u.  ( X 
\  A ) ) ) : ( ( X  i^i  A )  u.  ( X  \  A ) ) --> 2o 
->  F : X --> 2o ) )
2625adantld 278 . . 3  |-  ( ph  ->  ( ( F : X
--> ~P 1o  /\  ( F  |`  ( ( X  i^i  A )  u.  ( X  \  A
) ) ) : ( ( X  i^i  A )  u.  ( X 
\  A ) ) --> 2o )  ->  F : X --> 2o ) )
2726anim1d 336 . 2  |-  ( ph  ->  ( ( ( F : X --> ~P 1o  /\  ( F  |`  (
( X  i^i  A
)  u.  ( X 
\  A ) ) ) : ( ( X  i^i  A )  u.  ( X  \  A ) ) --> 2o )  /\  ( A. x  e.  ( X  i^i  A ) ( F `
 x )  =  1o  /\  A. x  e.  ( X  \  A
) ( F `  x )  =  (/) ) )  ->  ( F : X --> 2o  /\  ( A. x  e.  ( X  i^i  A ) ( F `  x
)  =  1o  /\  A. x  e.  ( X 
\  A ) ( F `  x )  =  (/) ) ) ) )
282, 27mpd 13 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 104  DECID wdc 835    = wceq 1364    e. wcel 2164   A.wral 2472    \ cdif 3151    u. cun 3152    i^i cin 3153    C_ wss 3154   (/)c0 3447   ifcif 3558   ~Pcpw 3602    |-> cmpt 4091    |` cres 4662   -->wf 5251   ` cfv 5255   1oc1o 6464   2oc2o 6465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-1o 6471  df-2o 6472
This theorem is referenced by: (None)
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