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Theorem rdgisuc1 6131
Description: One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function  F other than  F  Fn  _V. Given that, the resulting expression encompasses both the expected successor term  ( F `  ( rec ( F ,  A ) `  B
) ) but also terms that correspond to the initial value  A and to limit ordinals  U_ x  e.  B ( F `  ( rec ( F ,  A ) `  x
) ).

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6132. (Contributed by Jim Kingdon, 9-Jun-2019.)

Hypotheses
Ref Expression
rdgisuc1.1  |-  ( ph  ->  F  Fn  _V )
rdgisuc1.2  |-  ( ph  ->  A  e.  V )
rdgisuc1.3  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
rdgisuc1  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) ) )
Distinct variable groups:    x, F    x, A    x, B    x, V
Allowed substitution hint:    ph( x)

Proof of Theorem rdgisuc1
StepHypRef Expression
1 rdgisuc1.1 . . 3  |-  ( ph  ->  F  Fn  _V )
2 rdgisuc1.2 . . 3  |-  ( ph  ->  A  e.  V )
3 rdgisuc1.3 . . . 4  |-  ( ph  ->  B  e.  On )
4 suceloni 4308 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
53, 4syl 14 . . 3  |-  ( ph  ->  suc  B  e.  On )
6 rdgival 6129 . . 3  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  suc  B  e.  On )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  U_ x  e.  suc  B ( F `  ( rec ( F ,  A
) `  x )
) ) )
71, 2, 5, 6syl3anc 1174 . 2  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) ) ) )
8 df-suc 4189 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
9 iuneq1 3738 . . . . . . 7  |-  ( suc 
B  =  ( B  u.  { B }
)  ->  U_ x  e. 
suc  B ( F `
 ( rec ( F ,  A ) `  x ) )  = 
U_ x  e.  ( B  u.  { B } ) ( F `
 ( rec ( F ,  A ) `  x ) ) )
108, 9ax-mp 7 . . . . . 6  |-  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) )  = 
U_ x  e.  ( B  u.  { B } ) ( F `
 ( rec ( F ,  A ) `  x ) )
11 iunxun 3804 . . . . . 6  |-  U_ x  e.  ( B  u.  { B } ) ( F `
 ( rec ( F ,  A ) `  x ) )  =  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x
) )  u.  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `  x
) ) )
1210, 11eqtri 2108 . . . . 5  |-  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) )  =  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x
) )  u.  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `  x
) ) )
13 fveq2 5289 . . . . . . . 8  |-  ( x  =  B  ->  ( rec ( F ,  A
) `  x )  =  ( rec ( F ,  A ) `  B ) )
1413fveq2d 5293 . . . . . . 7  |-  ( x  =  B  ->  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1514iunxsng 3800 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1615uneq2d 3152 . . . . 5  |-  ( B  e.  On  ->  ( U_ x  e.  B  ( F `  ( rec ( F ,  A
) `  x )
)  u.  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `
 x ) ) )  =  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) )
1712, 16syl5eq 2132 . . . 4  |-  ( B  e.  On  ->  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) )  =  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x
) )  u.  ( F `  ( rec ( F ,  A ) `
 B ) ) ) )
1817uneq2d 3152 . . 3  |-  ( B  e.  On  ->  ( A  u.  U_ x  e. 
suc  B ( F `
 ( rec ( F ,  A ) `  x ) ) )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A
) `  x )
)  u.  ( F `
 ( rec ( F ,  A ) `  B ) ) ) ) )
193, 18syl 14 . 2  |-  ( ph  ->  ( A  u.  U_ x  e.  suc  B ( F `  ( rec ( F ,  A
) `  x )
) )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) ) )
207, 19eqtrd 2120 1  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619    u. cun 2995   {csn 3441   U_ciun 3725   Oncon0 4181   suc csuc 4183    Fn wfn 4997   ` cfv 5002   reccrdg 6116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-recs 6052  df-irdg 6117
This theorem is referenced by:  rdgisucinc  6132
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