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Theorem rdgisucinc 6003
Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6075 and omsuc 6082. (Contributed by Jim Kingdon, 29-Aug-2019.)

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

Proof of Theorem rdgisucinc
StepHypRef Expression
1 rdgisuc1.1 . . . 4  |-  ( ph  ->  F  Fn  _V )
2 rdgisuc1.2 . . . 4  |-  ( ph  ->  A  e.  V )
3 rdgisuc1.3 . . . 4  |-  ( ph  ->  B  e.  On )
41, 2, 3rdgisuc1 6002 . . 3  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) ) )
5 unass 3128 . . 3  |-  ( ( A  u.  U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x ) ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A
) `  x )
)  u.  ( F `
 ( rec ( F ,  A ) `  B ) ) ) )
64, 5syl6eqr 2106 . 2  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( ( A  u.  U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) ) )  u.  ( F `
 ( rec ( F ,  A ) `  B ) ) ) )
7 rdgival 6000 . . . 4  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  =  ( A  u.  U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x ) ) ) )
81, 2, 3, 7syl3anc 1146 . . 3  |-  ( ph  ->  ( rec ( F ,  A ) `  B )  =  ( A  u.  U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x ) ) ) )
98uneq1d 3124 . 2  |-  ( ph  ->  ( ( rec ( F ,  A ) `  B )  u.  ( F `  ( rec ( F ,  A ) `
 B ) ) )  =  ( ( A  u.  U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x ) ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) )
10 rdgexggg 5995 . . . . 5  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  e.  _V )
111, 2, 3, 10syl3anc 1146 . . . 4  |-  ( ph  ->  ( rec ( F ,  A ) `  B )  e.  _V )
12 rdgisucinc.inc . . . 4  |-  ( ph  ->  A. x  x  C_  ( F `  x ) )
13 id 19 . . . . . 6  |-  ( x  =  ( rec ( F ,  A ) `  B )  ->  x  =  ( rec ( F ,  A ) `  B ) )
14 fveq2 5206 . . . . . 6  |-  ( x  =  ( rec ( F ,  A ) `  B )  ->  ( F `  x )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1513, 14sseq12d 3002 . . . . 5  |-  ( x  =  ( rec ( F ,  A ) `  B )  ->  (
x  C_  ( F `  x )  <->  ( rec ( F ,  A ) `
 B )  C_  ( F `  ( rec ( F ,  A
) `  B )
) ) )
1615spcgv 2657 . . . 4  |-  ( ( rec ( F ,  A ) `  B
)  e.  _V  ->  ( A. x  x  C_  ( F `  x )  ->  ( rec ( F ,  A ) `  B )  C_  ( F `  ( rec ( F ,  A ) `
 B ) ) ) )
1711, 12, 16sylc 60 . . 3  |-  ( ph  ->  ( rec ( F ,  A ) `  B )  C_  ( F `  ( rec ( F ,  A ) `
 B ) ) )
18 ssequn1 3141 . . 3  |-  ( ( rec ( F ,  A ) `  B
)  C_  ( F `  ( rec ( F ,  A ) `  B ) )  <->  ( ( rec ( F ,  A
) `  B )  u.  ( F `  ( rec ( F ,  A
) `  B )
) )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
1917, 18sylib 131 . 2  |-  ( ph  ->  ( ( rec ( F ,  A ) `  B )  u.  ( F `  ( rec ( F ,  A ) `
 B ) ) )  =  ( F `
 ( rec ( F ,  A ) `  B ) ) )
206, 9, 193eqtr2d 2094 1  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259    e. wcel 1409   _Vcvv 2574    u. cun 2943    C_ wss 2945   U_ciun 3685   Oncon0 4128   suc csuc 4130    Fn wfn 4925   ` cfv 4930   reccrdg 5987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-recs 5951  df-irdg 5988
This theorem is referenced by:  frecrdg  6023
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