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Theorem frecrdg 6108
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6091 produces the same results as df-irdg 6070 restricted to  om.

Presumably the theorem would also hold if  F  Fn  _V were changed to  A. z ( F `  z )  e.  _V. (Contributed by Jim Kingdon, 29-Aug-2019.)

Hypotheses
Ref Expression
frecrdg.1  |-  ( ph  ->  F  Fn  _V )
frecrdg.2  |-  ( ph  ->  A  e.  V )
frecrdg.inc  |-  ( ph  ->  A. x  x  C_  ( F `  x ) )
Assertion
Ref Expression
frecrdg  |-  ( ph  -> frec ( F ,  A
)  =  ( rec ( F ,  A
)  |`  om ) )
Distinct variable groups:    x, A    x, F    x, V    ph, x

Proof of Theorem frecrdg
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecrdg.1 . . . 4  |-  ( ph  ->  F  Fn  _V )
2 vex 2617 . . . . . 6  |-  z  e. 
_V
3 funfvex 5270 . . . . . . 7  |-  ( ( Fun  F  /\  z  e.  dom  F )  -> 
( F `  z
)  e.  _V )
43funfni 5070 . . . . . 6  |-  ( ( F  Fn  _V  /\  z  e.  _V )  ->  ( F `  z
)  e.  _V )
52, 4mpan2 416 . . . . 5  |-  ( F  Fn  _V  ->  ( F `  z )  e.  _V )
65alrimiv 1799 . . . 4  |-  ( F  Fn  _V  ->  A. z
( F `  z
)  e.  _V )
71, 6syl 14 . . 3  |-  ( ph  ->  A. z ( F `
 z )  e. 
_V )
8 frecrdg.2 . . 3  |-  ( ph  ->  A  e.  V )
9 frecfnom 6101 . . 3  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  -> frec ( F ,  A )  Fn 
om )
107, 8, 9syl2anc 403 . 2  |-  ( ph  -> frec ( F ,  A
)  Fn  om )
11 rdgifnon2 6080 . . . 4  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
127, 8, 11syl2anc 403 . . 3  |-  ( ph  ->  rec ( F ,  A )  Fn  On )
13 omsson 4393 . . 3  |-  om  C_  On
14 fnssres 5083 . . 3  |-  ( ( rec ( F ,  A )  Fn  On  /\ 
om  C_  On )  -> 
( rec ( F ,  A )  |`  om )  Fn  om )
1512, 13, 14sylancl 404 . 2  |-  ( ph  ->  ( rec ( F ,  A )  |`  om )  Fn  om )
16 fveq2 5256 . . . . 5  |-  ( x  =  (/)  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  (/) ) )
17 fveq2 5256 . . . . 5  |-  ( x  =  (/)  ->  ( ( rec ( F ,  A )  |`  om ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  (/) ) )
1816, 17eqeq12d 2099 . . . 4  |-  ( x  =  (/)  ->  ( (frec ( F ,  A
) `  x )  =  ( ( rec ( F ,  A
)  |`  om ) `  x )  <->  (frec ( F ,  A ) `  (/) )  =  ( ( rec ( F ,  A )  |`  om ) `  (/) ) ) )
19 fveq2 5256 . . . . 5  |-  ( x  =  y  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  y
) )
20 fveq2 5256 . . . . 5  |-  ( x  =  y  ->  (
( rec ( F ,  A )  |`  om ) `  x )  =  ( ( rec ( F ,  A
)  |`  om ) `  y ) )
2119, 20eqeq12d 2099 . . . 4  |-  ( x  =  y  ->  (
(frec ( F ,  A ) `  x
)  =  ( ( rec ( F ,  A )  |`  om ) `  x )  <->  (frec ( F ,  A ) `  y )  =  ( ( rec ( F ,  A )  |`  om ) `  y ) ) )
22 fveq2 5256 . . . . 5  |-  ( x  =  suc  y  -> 
(frec ( F ,  A ) `  x
)  =  (frec ( F ,  A ) `
 suc  y )
)
23 fveq2 5256 . . . . 5  |-  ( x  =  suc  y  -> 
( ( rec ( F ,  A )  |` 
om ) `  x
)  =  ( ( rec ( F ,  A )  |`  om ) `  suc  y ) )
2422, 23eqeq12d 2099 . . . 4  |-  ( x  =  suc  y  -> 
( (frec ( F ,  A ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  x )  <-> 
(frec ( F ,  A ) `  suc  y )  =  ( ( rec ( F ,  A )  |`  om ) `  suc  y
) ) )
25 frec0g 6097 . . . . . 6  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )
268, 25syl 14 . . . . 5  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  =  A )
27 peano1 4375 . . . . . . 7  |-  (/)  e.  om
28 fvres 5277 . . . . . . 7  |-  ( (/)  e.  om  ->  ( ( rec ( F ,  A
)  |`  om ) `  (/) )  =  ( rec ( F ,  A
) `  (/) ) )
2927, 28ax-mp 7 . . . . . 6  |-  ( ( rec ( F ,  A )  |`  om ) `  (/) )  =  ( rec ( F ,  A ) `  (/) )
30 rdg0g 6088 . . . . . . 7  |-  ( A  e.  V  ->  ( rec ( F ,  A
) `  (/) )  =  A )
318, 30syl 14 . . . . . 6  |-  ( ph  ->  ( rec ( F ,  A ) `  (/) )  =  A )
3229, 31syl5eq 2129 . . . . 5  |-  ( ph  ->  ( ( rec ( F ,  A )  |` 
om ) `  (/) )  =  A )
3326, 32eqtr4d 2120 . . . 4  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  =  ( ( rec ( F ,  A )  |` 
om ) `  (/) ) )
34 simpr 108 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )
35 fvres 5277 . . . . . . . . . . 11  |-  ( y  e.  om  ->  (
( rec ( F ,  A )  |`  om ) `  y )  =  ( rec ( F ,  A ) `  y ) )
3635ad2antlr 473 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (
( rec ( F ,  A )  |`  om ) `  y )  =  ( rec ( F ,  A ) `  y ) )
3734, 36eqtrd 2117 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 y )  =  ( rec ( F ,  A ) `  y ) )
3837fveq2d 5260 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  ( F `  (frec ( F ,  A ) `  y ) )  =  ( F `  ( rec ( F ,  A
) `  y )
) )
397, 8jca 300 . . . . . . . . . 10  |-  ( ph  ->  ( A. z ( F `  z )  e.  _V  /\  A  e.  V ) )
40 simp1 941 . . . . . . . . . . . . 13  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A. z ( F `
 z )  e. 
_V )
41 ralv 2629 . . . . . . . . . . . . 13  |-  ( A. z  e.  _V  ( F `  z )  e.  _V  <->  A. z ( F `
 z )  e. 
_V )
4240, 41sylibr 132 . . . . . . . . . . . 12  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A. z  e.  _V  ( F `  z )  e.  _V )
43 simp2 942 . . . . . . . . . . . . 13  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A  e.  V
)
4443elexd 2625 . . . . . . . . . . . 12  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A  e.  _V )
45 simp3 943 . . . . . . . . . . . 12  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  y  e.  om )
46 frecsuc 6107 . . . . . . . . . . . 12  |-  ( ( A. z  e.  _V  ( F `  z )  e.  _V  /\  A  e.  _V  /\  y  e. 
om )  ->  (frec ( F ,  A ) `
 suc  y )  =  ( F `  (frec ( F ,  A
) `  y )
) )
4742, 44, 45, 46syl3anc 1172 . . . . . . . . . . 11  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
48473expa 1141 . . . . . . . . . 10  |-  ( ( ( A. z ( F `  z )  e.  _V  /\  A  e.  V )  /\  y  e.  om )  ->  (frec ( F ,  A ) `
 suc  y )  =  ( F `  (frec ( F ,  A
) `  y )
) )
4939, 48sylan 277 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
5049adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 suc  y )  =  ( F `  (frec ( F ,  A
) `  y )
) )
511adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  F  Fn  _V )
528adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  A  e.  V )
53 simpr 108 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  om )  ->  y  e.  om )
54 nnon 4390 . . . . . . . . . . 11  |-  ( y  e.  om  ->  y  e.  On )
5553, 54syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  y  e.  On )
56 frecrdg.inc . . . . . . . . . . 11  |-  ( ph  ->  A. x  x  C_  ( F `  x ) )
5756adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  A. x  x  C_  ( F `  x ) )
5851, 52, 55, 57rdgisucinc 6085 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  om )  ->  ( rec ( F ,  A ) `
 suc  y )  =  ( F `  ( rec ( F ,  A ) `  y
) ) )
5958adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  ( rec ( F ,  A
) `  suc  y )  =  ( F `  ( rec ( F ,  A ) `  y
) ) )
6038, 50, 593eqtr4d 2127 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 suc  y )  =  ( rec ( F ,  A ) `  suc  y ) )
61 peano2 4376 . . . . . . . . 9  |-  ( y  e.  om  ->  suc  y  e.  om )
62 fvres 5277 . . . . . . . . 9  |-  ( suc  y  e.  om  ->  ( ( rec ( F ,  A )  |`  om ) `  suc  y
)  =  ( rec ( F ,  A
) `  suc  y ) )
6361, 62syl 14 . . . . . . . 8  |-  ( y  e.  om  ->  (
( rec ( F ,  A )  |`  om ) `  suc  y
)  =  ( rec ( F ,  A
) `  suc  y ) )
6463ad2antlr 473 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (
( rec ( F ,  A )  |`  om ) `  suc  y
)  =  ( rec ( F ,  A
) `  suc  y ) )
6560, 64eqtr4d 2120 . . . . . 6  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 suc  y )  =  ( ( rec ( F ,  A
)  |`  om ) `  suc  y ) )
6665ex 113 . . . . 5  |-  ( (
ph  /\  y  e.  om )  ->  ( (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
)  ->  (frec ( F ,  A ) `  suc  y )  =  ( ( rec ( F ,  A )  |` 
om ) `  suc  y ) ) )
6766expcom 114 . . . 4  |-  ( y  e.  om  ->  ( ph  ->  ( (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
)  ->  (frec ( F ,  A ) `  suc  y )  =  ( ( rec ( F ,  A )  |` 
om ) `  suc  y ) ) ) )
6818, 21, 24, 33, 67finds2 4382 . . 3  |-  ( x  e.  om  ->  ( ph  ->  (frec ( F ,  A ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  x ) ) )
6968impcom 123 . 2  |-  ( (
ph  /\  x  e.  om )  ->  (frec ( F ,  A ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  x ) )
7010, 15, 69eqfnfvd 5348 1  |-  ( ph  -> frec ( F ,  A
)  =  ( rec ( F ,  A
)  |`  om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 922   A.wal 1285    = wceq 1287    e. wcel 1436   A.wral 2355   _Vcvv 2614    C_ wss 2986   (/)c0 3272   Oncon0 4157   suc csuc 4159   omcom 4371    |` cres 4406    Fn wfn 4967   ` cfv 4972   reccrdg 6069  freccfrec 6090
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-recs 6005  df-irdg 6070  df-frec 6091
This theorem is referenced by: (None)
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