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Theorem frecrdg 6311
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6294 produces the same results as df-irdg 6273 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 2692 . . . . . 6  |-  z  e. 
_V
3 funfvex 5444 . . . . . . 7  |-  ( ( Fun  F  /\  z  e.  dom  F )  -> 
( F `  z
)  e.  _V )
43funfni 5229 . . . . . 6  |-  ( ( F  Fn  _V  /\  z  e.  _V )  ->  ( F `  z
)  e.  _V )
52, 4mpan2 422 . . . . 5  |-  ( F  Fn  _V  ->  ( F `  z )  e.  _V )
65alrimiv 1847 . . . 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 6304 . . 3  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  -> frec ( F ,  A )  Fn 
om )
107, 8, 9syl2anc 409 . 2  |-  ( ph  -> frec ( F ,  A
)  Fn  om )
11 rdgifnon2 6283 . . . 4  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
127, 8, 11syl2anc 409 . . 3  |-  ( ph  ->  rec ( F ,  A )  Fn  On )
13 omsson 4532 . . 3  |-  om  C_  On
14 fnssres 5242 . . 3  |-  ( ( rec ( F ,  A )  Fn  On  /\ 
om  C_  On )  -> 
( rec ( F ,  A )  |`  om )  Fn  om )
1512, 13, 14sylancl 410 . 2  |-  ( ph  ->  ( rec ( F ,  A )  |`  om )  Fn  om )
16 fveq2 5427 . . . . 5  |-  ( x  =  (/)  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  (/) ) )
17 fveq2 5427 . . . . 5  |-  ( x  =  (/)  ->  ( ( rec ( F ,  A )  |`  om ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  (/) ) )
1816, 17eqeq12d 2155 . . . 4  |-  ( x  =  (/)  ->  ( (frec ( F ,  A
) `  x )  =  ( ( rec ( F ,  A
)  |`  om ) `  x )  <->  (frec ( F ,  A ) `  (/) )  =  ( ( rec ( F ,  A )  |`  om ) `  (/) ) ) )
19 fveq2 5427 . . . . 5  |-  ( x  =  y  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  y
) )
20 fveq2 5427 . . . . 5  |-  ( x  =  y  ->  (
( rec ( F ,  A )  |`  om ) `  x )  =  ( ( rec ( F ,  A
)  |`  om ) `  y ) )
2119, 20eqeq12d 2155 . . . 4  |-  ( x  =  y  ->  (
(frec ( F ,  A ) `  x
)  =  ( ( rec ( F ,  A )  |`  om ) `  x )  <->  (frec ( F ,  A ) `  y )  =  ( ( rec ( F ,  A )  |`  om ) `  y ) ) )
22 fveq2 5427 . . . . 5  |-  ( x  =  suc  y  -> 
(frec ( F ,  A ) `  x
)  =  (frec ( F ,  A ) `
 suc  y )
)
23 fveq2 5427 . . . . 5  |-  ( x  =  suc  y  -> 
( ( rec ( F ,  A )  |` 
om ) `  x
)  =  ( ( rec ( F ,  A )  |`  om ) `  suc  y ) )
2422, 23eqeq12d 2155 . . . 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 6300 . . . . . 6  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )
268, 25syl 14 . . . . 5  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  =  A )
27 peano1 4514 . . . . . . 7  |-  (/)  e.  om
28 fvres 5451 . . . . . . 7  |-  ( (/)  e.  om  ->  ( ( rec ( F ,  A
)  |`  om ) `  (/) )  =  ( rec ( F ,  A
) `  (/) ) )
2927, 28ax-mp 5 . . . . . 6  |-  ( ( rec ( F ,  A )  |`  om ) `  (/) )  =  ( rec ( F ,  A ) `  (/) )
30 rdg0g 6291 . . . . . . 7  |-  ( A  e.  V  ->  ( rec ( F ,  A
) `  (/) )  =  A )
318, 30syl 14 . . . . . 6  |-  ( ph  ->  ( rec ( F ,  A ) `  (/) )  =  A )
3229, 31syl5eq 2185 . . . . 5  |-  ( ph  ->  ( ( rec ( F ,  A )  |` 
om ) `  (/) )  =  A )
3326, 32eqtr4d 2176 . . . 4  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  =  ( ( rec ( F ,  A )  |` 
om ) `  (/) ) )
34 simpr 109 . . . . . . . . . 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 5451 . . . . . . . . . . 11  |-  ( y  e.  om  ->  (
( rec ( F ,  A )  |`  om ) `  y )  =  ( rec ( F ,  A ) `  y ) )
3635ad2antlr 481 . . . . . . . . . 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 2173 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 y )  =  ( rec ( F ,  A ) `  y ) )
3837fveq2d 5431 . . . . . . . 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 304 . . . . . . . . . 10  |-  ( ph  ->  ( A. z ( F `  z )  e.  _V  /\  A  e.  V ) )
40 simp1 982 . . . . . . . . . . . . 13  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A. z ( F `
 z )  e. 
_V )
41 ralv 2706 . . . . . . . . . . . . 13  |-  ( A. z  e.  _V  ( F `  z )  e.  _V  <->  A. z ( F `
 z )  e. 
_V )
4240, 41sylibr 133 . . . . . . . . . . . 12  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A. z  e.  _V  ( F `  z )  e.  _V )
43 simp2 983 . . . . . . . . . . . . 13  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A  e.  V
)
4443elexd 2702 . . . . . . . . . . . 12  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A  e.  _V )
45 simp3 984 . . . . . . . . . . . 12  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  y  e.  om )
46 frecsuc 6310 . . . . . . . . . . . 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 1217 . . . . . . . . . . 11  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
48473expa 1182 . . . . . . . . . 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 281 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
5049adantr 274 . . . . . . . 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 274 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  F  Fn  _V )
528adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  A  e.  V )
53 simpr 109 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  om )  ->  y  e.  om )
54 nnon 4529 . . . . . . . . . . 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 274 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  om )  ->  A. x  x  C_  ( F `  x ) )
5851, 52, 55, 57rdgisucinc 6288 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  om )  ->  ( rec ( F ,  A ) `
 suc  y )  =  ( F `  ( rec ( F ,  A ) `  y
) ) )
5958adantr 274 . . . . . . . 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 2183 . . . . . . 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 4515 . . . . . . . . 9  |-  ( y  e.  om  ->  suc  y  e.  om )
62 fvres 5451 . . . . . . . . 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 481 . . . . . . 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 2176 . . . . . 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 114 . . . . 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 115 . . . 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 4521 . . 3  |-  ( x  e.  om  ->  ( ph  ->  (frec ( F ,  A ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  x ) ) )
6968impcom 124 . 2  |-  ( (
ph  /\  x  e.  om )  ->  (frec ( F ,  A ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  x ) )
7010, 15, 69eqfnfvd 5527 1  |-  ( ph  -> frec ( F ,  A
)  =  ( rec ( F ,  A
)  |`  om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963   A.wal 1330    = wceq 1332    e. wcel 1481   A.wral 2417   _Vcvv 2689    C_ wss 3074   (/)c0 3366   Oncon0 4291   suc csuc 4293   omcom 4510    |` cres 4547    Fn wfn 5124   ` cfv 5129   reccrdg 6272  freccfrec 6293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-iord 4294  df-on 4296  df-ilim 4297  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-recs 6208  df-irdg 6273  df-frec 6294
This theorem is referenced by: (None)
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