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Theorem freccl 6021
Description: Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)
Hypotheses
Ref Expression
freccl.ex  |-  ( ph  ->  A. z ( F `
 z )  e. 
_V )
freccl.a  |-  ( ph  ->  A  e.  S )
freccl.cl  |-  ( (
ph  /\  z  e.  S )  ->  ( F `  z )  e.  S )
freccl.b  |-  ( ph  ->  B  e.  om )
Assertion
Ref Expression
freccl  |-  ( ph  ->  (frec ( F ,  A ) `  B
)  e.  S )
Distinct variable groups:    z, A    z, F    z, S    ph, z
Allowed substitution hint:    B( z)

Proof of Theorem freccl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 freccl.b . 2  |-  ( ph  ->  B  e.  om )
2 fveq2 5203 . . . . 5  |-  ( x  =  B  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  B
) )
32eleq1d 2120 . . . 4  |-  ( x  =  B  ->  (
(frec ( F ,  A ) `  x
)  e.  S  <->  (frec ( F ,  A ) `  B )  e.  S
) )
43imbi2d 223 . . 3  |-  ( x  =  B  ->  (
( ph  ->  (frec ( F ,  A ) `
 x )  e.  S )  <->  ( ph  ->  (frec ( F ,  A ) `  B
)  e.  S ) ) )
5 fveq2 5203 . . . . 5  |-  ( x  =  (/)  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  (/) ) )
65eleq1d 2120 . . . 4  |-  ( x  =  (/)  ->  ( (frec ( F ,  A
) `  x )  e.  S  <->  (frec ( F ,  A ) `  (/) )  e.  S ) )
7 fveq2 5203 . . . . 5  |-  ( x  =  y  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  y
) )
87eleq1d 2120 . . . 4  |-  ( x  =  y  ->  (
(frec ( F ,  A ) `  x
)  e.  S  <->  (frec ( F ,  A ) `  y )  e.  S
) )
9 fveq2 5203 . . . . 5  |-  ( x  =  suc  y  -> 
(frec ( F ,  A ) `  x
)  =  (frec ( F ,  A ) `
 suc  y )
)
109eleq1d 2120 . . . 4  |-  ( x  =  suc  y  -> 
( (frec ( F ,  A ) `  x )  e.  S  <->  (frec ( F ,  A
) `  suc  y )  e.  S ) )
11 freccl.a . . . . . 6  |-  ( ph  ->  A  e.  S )
12 frec0g 6011 . . . . . 6  |-  ( A  e.  S  ->  (frec ( F ,  A ) `
 (/) )  =  A )
1311, 12syl 14 . . . . 5  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  =  A )
1413, 11eqeltrd 2128 . . . 4  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  e.  S )
15 freccl.ex . . . . . . . . . 10  |-  ( ph  ->  A. z ( F `
 z )  e. 
_V )
16 frecfnom 6014 . . . . . . . . . 10  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  S )  -> frec ( F ,  A )  Fn 
om )
1715, 11, 16syl2anc 397 . . . . . . . . 9  |-  ( ph  -> frec ( F ,  A
)  Fn  om )
18 funfvex 5217 . . . . . . . . . 10  |-  ( ( Fun frec ( F ,  A )  /\  y  e.  dom frec ( F ,  A ) )  -> 
(frec ( F ,  A ) `  y
)  e.  _V )
1918funfni 5024 . . . . . . . . 9  |-  ( (frec ( F ,  A
)  Fn  om  /\  y  e.  om )  ->  (frec ( F ,  A ) `  y
)  e.  _V )
2017, 19sylan 271 . . . . . . . 8  |-  ( (
ph  /\  y  e.  om )  ->  (frec ( F ,  A ) `  y )  e.  _V )
21 isset 2576 . . . . . . . 8  |-  ( (frec ( F ,  A
) `  y )  e.  _V  <->  E. z  z  =  (frec ( F ,  A ) `  y
) )
2220, 21sylib 131 . . . . . . 7  |-  ( (
ph  /\  y  e.  om )  ->  E. z 
z  =  (frec ( F ,  A ) `
 y ) )
23 freccl.cl . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  S )  ->  ( F `  z )  e.  S )
2423ex 112 . . . . . . . . . . . 12  |-  ( ph  ->  ( z  e.  S  ->  ( F `  z
)  e.  S ) )
2524adantr 265 . . . . . . . . . . 11  |-  ( (
ph  /\  z  =  (frec ( F ,  A
) `  y )
)  ->  ( z  e.  S  ->  ( F `
 z )  e.  S ) )
26 eleq1 2114 . . . . . . . . . . . 12  |-  ( z  =  (frec ( F ,  A ) `  y )  ->  (
z  e.  S  <->  (frec ( F ,  A ) `  y )  e.  S
) )
2726adantl 266 . . . . . . . . . . 11  |-  ( (
ph  /\  z  =  (frec ( F ,  A
) `  y )
)  ->  ( z  e.  S  <->  (frec ( F ,  A ) `  y
)  e.  S ) )
28 fveq2 5203 . . . . . . . . . . . . 13  |-  ( z  =  (frec ( F ,  A ) `  y )  ->  ( F `  z )  =  ( F `  (frec ( F ,  A
) `  y )
) )
2928eleq1d 2120 . . . . . . . . . . . 12  |-  ( z  =  (frec ( F ,  A ) `  y )  ->  (
( F `  z
)  e.  S  <->  ( F `  (frec ( F ,  A ) `  y
) )  e.  S
) )
3029adantl 266 . . . . . . . . . . 11  |-  ( (
ph  /\  z  =  (frec ( F ,  A
) `  y )
)  ->  ( ( F `  z )  e.  S  <->  ( F `  (frec ( F ,  A
) `  y )
)  e.  S ) )
3125, 27, 303imtr3d 195 . . . . . . . . . 10  |-  ( (
ph  /\  z  =  (frec ( F ,  A
) `  y )
)  ->  ( (frec ( F ,  A ) `
 y )  e.  S  ->  ( F `  (frec ( F ,  A ) `  y
) )  e.  S
) )
3231ex 112 . . . . . . . . 9  |-  ( ph  ->  ( z  =  (frec ( F ,  A
) `  y )  ->  ( (frec ( F ,  A ) `  y )  e.  S  ->  ( F `  (frec ( F ,  A ) `
 y ) )  e.  S ) ) )
3332exlimdv 1714 . . . . . . . 8  |-  ( ph  ->  ( E. z  z  =  (frec ( F ,  A ) `  y )  ->  (
(frec ( F ,  A ) `  y
)  e.  S  -> 
( F `  (frec ( F ,  A ) `
 y ) )  e.  S ) ) )
3433adantr 265 . . . . . . 7  |-  ( (
ph  /\  y  e.  om )  ->  ( E. z  z  =  (frec ( F ,  A ) `
 y )  -> 
( (frec ( F ,  A ) `  y )  e.  S  ->  ( F `  (frec ( F ,  A ) `
 y ) )  e.  S ) ) )
3522, 34mpd 13 . . . . . 6  |-  ( (
ph  /\  y  e.  om )  ->  ( (frec ( F ,  A ) `
 y )  e.  S  ->  ( F `  (frec ( F ,  A ) `  y
) )  e.  S
) )
3615adantr 265 . . . . . . . 8  |-  ( (
ph  /\  y  e.  om )  ->  A. z
( F `  z
)  e.  _V )
3711adantr 265 . . . . . . . 8  |-  ( (
ph  /\  y  e.  om )  ->  A  e.  S )
38 simpr 107 . . . . . . . 8  |-  ( (
ph  /\  y  e.  om )  ->  y  e.  om )
39 frecsuc 6019 . . . . . . . 8  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  S  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
4036, 37, 38, 39syl3anc 1144 . . . . . . 7  |-  ( (
ph  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
4140eleq1d 2120 . . . . . 6  |-  ( (
ph  /\  y  e.  om )  ->  ( (frec ( F ,  A ) `
 suc  y )  e.  S  <->  ( F `  (frec ( F ,  A
) `  y )
)  e.  S ) )
4235, 41sylibrd 162 . . . . 5  |-  ( (
ph  /\  y  e.  om )  ->  ( (frec ( F ,  A ) `
 y )  e.  S  ->  (frec ( F ,  A ) `  suc  y )  e.  S ) )
4342expcom 113 . . . 4  |-  ( y  e.  om  ->  ( ph  ->  ( (frec ( F ,  A ) `
 y )  e.  S  ->  (frec ( F ,  A ) `  suc  y )  e.  S ) ) )
446, 8, 10, 14, 43finds2 4349 . . 3  |-  ( x  e.  om  ->  ( ph  ->  (frec ( F ,  A ) `  x )  e.  S
) )
454, 44vtoclga 2634 . 2  |-  ( B  e.  om  ->  ( ph  ->  (frec ( F ,  A ) `  B )  e.  S
) )
461, 45mpcom 36 1  |-  ( ph  ->  (frec ( F ,  A ) `  B
)  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1255    = wceq 1257   E.wex 1395    e. wcel 1407   _Vcvv 2572   (/)c0 3249   suc csuc 4127   omcom 4338    Fn wfn 4922   ` cfv 4927  freccfrec 6005
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 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-id 4055  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-recs 5948  df-frec 6006
This theorem is referenced by:  frecuzrdgrrn  9323
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