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Theorem frecrdg 5992
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 5978 produces the same results as df-irdg 5957 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 2560 . . . . . 6  |-  z  e. 
_V
3 funfvex 5192 . . . . . . 7  |-  ( ( Fun  F  /\  z  e.  dom  F )  -> 
( F `  z
)  e.  _V )
43funfni 4999 . . . . . 6  |-  ( ( F  Fn  _V  /\  z  e.  _V )  ->  ( F `  z
)  e.  _V )
52, 4mpan2 401 . . . . 5  |-  ( F  Fn  _V  ->  ( F `  z )  e.  _V )
65alrimiv 1754 . . . 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 5986 . . 3  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  -> frec ( F ,  A )  Fn 
om )
107, 8, 9syl2anc 391 . 2  |-  ( ph  -> frec ( F ,  A
)  Fn  om )
11 rdgifnon2 5967 . . . 4  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
127, 8, 11syl2anc 391 . . 3  |-  ( ph  ->  rec ( F ,  A )  Fn  On )
13 omsson 4335 . . 3  |-  om  C_  On
14 fnssres 5012 . . 3  |-  ( ( rec ( F ,  A )  Fn  On  /\ 
om  C_  On )  -> 
( rec ( F ,  A )  |`  om )  Fn  om )
1512, 13, 14sylancl 392 . 2  |-  ( ph  ->  ( rec ( F ,  A )  |`  om )  Fn  om )
16 fveq2 5178 . . . . 5  |-  ( x  =  (/)  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  (/) ) )
17 fveq2 5178 . . . . 5  |-  ( x  =  (/)  ->  ( ( rec ( F ,  A )  |`  om ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  (/) ) )
1816, 17eqeq12d 2054 . . . 4  |-  ( x  =  (/)  ->  ( (frec ( F ,  A
) `  x )  =  ( ( rec ( F ,  A
)  |`  om ) `  x )  <->  (frec ( F ,  A ) `  (/) )  =  ( ( rec ( F ,  A )  |`  om ) `  (/) ) ) )
19 fveq2 5178 . . . . 5  |-  ( x  =  y  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  y
) )
20 fveq2 5178 . . . . 5  |-  ( x  =  y  ->  (
( rec ( F ,  A )  |`  om ) `  x )  =  ( ( rec ( F ,  A
)  |`  om ) `  y ) )
2119, 20eqeq12d 2054 . . . 4  |-  ( x  =  y  ->  (
(frec ( F ,  A ) `  x
)  =  ( ( rec ( F ,  A )  |`  om ) `  x )  <->  (frec ( F ,  A ) `  y )  =  ( ( rec ( F ,  A )  |`  om ) `  y ) ) )
22 fveq2 5178 . . . . 5  |-  ( x  =  suc  y  -> 
(frec ( F ,  A ) `  x
)  =  (frec ( F ,  A ) `
 suc  y )
)
23 fveq2 5178 . . . . 5  |-  ( x  =  suc  y  -> 
( ( rec ( F ,  A )  |` 
om ) `  x
)  =  ( ( rec ( F ,  A )  |`  om ) `  suc  y ) )
2422, 23eqeq12d 2054 . . . 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 5983 . . . . . 6  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )
268, 25syl 14 . . . . 5  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  =  A )
27 peano1 4317 . . . . . . 7  |-  (/)  e.  om
28 fvres 5198 . . . . . . 7  |-  ( (/)  e.  om  ->  ( ( rec ( F ,  A
)  |`  om ) `  (/) )  =  ( rec ( F ,  A
) `  (/) ) )
2927, 28ax-mp 7 . . . . . 6  |-  ( ( rec ( F ,  A )  |`  om ) `  (/) )  =  ( rec ( F ,  A ) `  (/) )
30 rdg0g 5975 . . . . . . 7  |-  ( A  e.  V  ->  ( rec ( F ,  A
) `  (/) )  =  A )
318, 30syl 14 . . . . . 6  |-  ( ph  ->  ( rec ( F ,  A ) `  (/) )  =  A )
3229, 31syl5eq 2084 . . . . 5  |-  ( ph  ->  ( ( rec ( F ,  A )  |` 
om ) `  (/) )  =  A )
3326, 32eqtr4d 2075 . . . 4  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  =  ( ( rec ( F ,  A )  |` 
om ) `  (/) ) )
34 simpr 103 . . . . . . . . . 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 5198 . . . . . . . . . . 11  |-  ( y  e.  om  ->  (
( rec ( F ,  A )  |`  om ) `  y )  =  ( rec ( F ,  A ) `  y ) )
3635ad2antlr 458 . . . . . . . . . 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 2072 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 y )  =  ( rec ( F ,  A ) `  y ) )
3837fveq2d 5182 . . . . . . . 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 290 . . . . . . . . . 10  |-  ( ph  ->  ( A. z ( F `  z )  e.  _V  /\  A  e.  V ) )
40 frecsuc 5991 . . . . . . . . . . 11  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
41403expa 1104 . . . . . . . . . 10  |-  ( ( ( A. z ( F `  z )  e.  _V  /\  A  e.  V )  /\  y  e.  om )  ->  (frec ( F ,  A ) `
 suc  y )  =  ( F `  (frec ( F ,  A
) `  y )
) )
4239, 41sylan 267 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
4342adantr 261 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 suc  y )  =  ( F `  (frec ( F ,  A
) `  y )
) )
441adantr 261 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  F  Fn  _V )
458adantr 261 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  A  e.  V )
46 simpr 103 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  om )  ->  y  e.  om )
47 nnon 4332 . . . . . . . . . . 11  |-  ( y  e.  om  ->  y  e.  On )
4846, 47syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  y  e.  On )
49 frecrdg.inc . . . . . . . . . . 11  |-  ( ph  ->  A. x  x  C_  ( F `  x ) )
5049adantr 261 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  A. x  x  C_  ( F `  x ) )
5144, 45, 48, 50rdgisucinc 5972 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  om )  ->  ( rec ( F ,  A ) `
 suc  y )  =  ( F `  ( rec ( F ,  A ) `  y
) ) )
5251adantr 261 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  ( rec ( F ,  A
) `  suc  y )  =  ( F `  ( rec ( F ,  A ) `  y
) ) )
5338, 43, 523eqtr4d 2082 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 suc  y )  =  ( rec ( F ,  A ) `  suc  y ) )
54 peano2 4318 . . . . . . . . 9  |-  ( y  e.  om  ->  suc  y  e.  om )
55 fvres 5198 . . . . . . . . 9  |-  ( suc  y  e.  om  ->  ( ( rec ( F ,  A )  |`  om ) `  suc  y
)  =  ( rec ( F ,  A
) `  suc  y ) )
5654, 55syl 14 . . . . . . . 8  |-  ( y  e.  om  ->  (
( rec ( F ,  A )  |`  om ) `  suc  y
)  =  ( rec ( F ,  A
) `  suc  y ) )
5756ad2antlr 458 . . . . . . 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 ) )
5853, 57eqtr4d 2075 . . . . . 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 ) )
5958ex 108 . . . . 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 ) ) )
6059expcom 109 . . . 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 ) ) ) )
6118, 21, 24, 33, 60finds2 4324 . . 3  |-  ( x  e.  om  ->  ( ph  ->  (frec ( F ,  A ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  x ) ) )
6261impcom 116 . 2  |-  ( (
ph  /\  x  e.  om )  ->  (frec ( F ,  A ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  x ) )
6310, 15, 62eqfnfvd 5268 1  |-  ( ph  -> frec ( F ,  A
)  =  ( rec ( F ,  A
)  |`  om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   A.wal 1241    = wceq 1243    e. wcel 1393   _Vcvv 2557    C_ wss 2917   (/)c0 3224   Oncon0 4100   suc csuc 4102   omcom 4313    |` cres 4347    Fn wfn 4897   ` cfv 4902   reccrdg 5956  freccfrec 5977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-irdg 5957  df-frec 5978
This theorem is referenced by: (None)
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