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Theorem frecrdg 6273
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6256 produces the same results as df-irdg 6235 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 2663 . . . . . 6  |-  z  e. 
_V
3 funfvex 5406 . . . . . . 7  |-  ( ( Fun  F  /\  z  e.  dom  F )  -> 
( F `  z
)  e.  _V )
43funfni 5193 . . . . . 6  |-  ( ( F  Fn  _V  /\  z  e.  _V )  ->  ( F `  z
)  e.  _V )
52, 4mpan2 421 . . . . 5  |-  ( F  Fn  _V  ->  ( F `  z )  e.  _V )
65alrimiv 1830 . . . 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 6266 . . 3  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  -> frec ( F ,  A )  Fn 
om )
107, 8, 9syl2anc 408 . 2  |-  ( ph  -> frec ( F ,  A
)  Fn  om )
11 rdgifnon2 6245 . . . 4  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
127, 8, 11syl2anc 408 . . 3  |-  ( ph  ->  rec ( F ,  A )  Fn  On )
13 omsson 4496 . . 3  |-  om  C_  On
14 fnssres 5206 . . 3  |-  ( ( rec ( F ,  A )  Fn  On  /\ 
om  C_  On )  -> 
( rec ( F ,  A )  |`  om )  Fn  om )
1512, 13, 14sylancl 409 . 2  |-  ( ph  ->  ( rec ( F ,  A )  |`  om )  Fn  om )
16 fveq2 5389 . . . . 5  |-  ( x  =  (/)  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  (/) ) )
17 fveq2 5389 . . . . 5  |-  ( x  =  (/)  ->  ( ( rec ( F ,  A )  |`  om ) `  x )  =  ( ( rec ( F ,  A )  |`  om ) `  (/) ) )
1816, 17eqeq12d 2132 . . . 4  |-  ( x  =  (/)  ->  ( (frec ( F ,  A
) `  x )  =  ( ( rec ( F ,  A
)  |`  om ) `  x )  <->  (frec ( F ,  A ) `  (/) )  =  ( ( rec ( F ,  A )  |`  om ) `  (/) ) ) )
19 fveq2 5389 . . . . 5  |-  ( x  =  y  ->  (frec ( F ,  A ) `
 x )  =  (frec ( F ,  A ) `  y
) )
20 fveq2 5389 . . . . 5  |-  ( x  =  y  ->  (
( rec ( F ,  A )  |`  om ) `  x )  =  ( ( rec ( F ,  A
)  |`  om ) `  y ) )
2119, 20eqeq12d 2132 . . . 4  |-  ( x  =  y  ->  (
(frec ( F ,  A ) `  x
)  =  ( ( rec ( F ,  A )  |`  om ) `  x )  <->  (frec ( F ,  A ) `  y )  =  ( ( rec ( F ,  A )  |`  om ) `  y ) ) )
22 fveq2 5389 . . . . 5  |-  ( x  =  suc  y  -> 
(frec ( F ,  A ) `  x
)  =  (frec ( F ,  A ) `
 suc  y )
)
23 fveq2 5389 . . . . 5  |-  ( x  =  suc  y  -> 
( ( rec ( F ,  A )  |` 
om ) `  x
)  =  ( ( rec ( F ,  A )  |`  om ) `  suc  y ) )
2422, 23eqeq12d 2132 . . . 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 6262 . . . . . 6  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )
268, 25syl 14 . . . . 5  |-  ( ph  ->  (frec ( F ,  A ) `  (/) )  =  A )
27 peano1 4478 . . . . . . 7  |-  (/)  e.  om
28 fvres 5413 . . . . . . 7  |-  ( (/)  e.  om  ->  ( ( rec ( F ,  A
)  |`  om ) `  (/) )  =  ( rec ( F ,  A
) `  (/) ) )
2927, 28ax-mp 5 . . . . . 6  |-  ( ( rec ( F ,  A )  |`  om ) `  (/) )  =  ( rec ( F ,  A ) `  (/) )
30 rdg0g 6253 . . . . . . 7  |-  ( A  e.  V  ->  ( rec ( F ,  A
) `  (/) )  =  A )
318, 30syl 14 . . . . . 6  |-  ( ph  ->  ( rec ( F ,  A ) `  (/) )  =  A )
3229, 31syl5eq 2162 . . . . 5  |-  ( ph  ->  ( ( rec ( F ,  A )  |` 
om ) `  (/) )  =  A )
3326, 32eqtr4d 2153 . . . 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 5413 . . . . . . . . . . 11  |-  ( y  e.  om  ->  (
( rec ( F ,  A )  |`  om ) `  y )  =  ( rec ( F ,  A ) `  y ) )
3635ad2antlr 480 . . . . . . . . . 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 2150 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  om )  /\  (frec ( F ,  A ) `
 y )  =  ( ( rec ( F ,  A )  |` 
om ) `  y
) )  ->  (frec ( F ,  A ) `
 y )  =  ( rec ( F ,  A ) `  y ) )
3837fveq2d 5393 . . . . . . . 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 966 . . . . . . . . . . . . 13  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A. z ( F `
 z )  e. 
_V )
41 ralv 2677 . . . . . . . . . . . . 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 967 . . . . . . . . . . . . 13  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A  e.  V
)
4443elexd 2673 . . . . . . . . . . . 12  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  A  e.  _V )
45 simp3 968 . . . . . . . . . . . 12  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  y  e.  om )
46 frecsuc 6272 . . . . . . . . . . . 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 1201 . . . . . . . . . . 11  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  y  e.  om )  ->  (frec ( F ,  A ) `  suc  y )  =  ( F `  (frec ( F ,  A ) `
 y ) ) )
48473expa 1166 . . . . . . . . . 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 4493 . . . . . . . . . . 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 6250 . . . . . . . . 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 2160 . . . . . . 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 4479 . . . . . . . . 9  |-  ( y  e.  om  ->  suc  y  e.  om )
62 fvres 5413 . . . . . . . . 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 480 . . . . . . 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 2153 . . . . . 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 4485 . . 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 5489 1  |-  ( ph  -> frec ( F ,  A
)  =  ( rec ( F ,  A
)  |`  om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393   _Vcvv 2660    C_ wss 3041   (/)c0 3333   Oncon0 4255   suc csuc 4257   omcom 4474    |` cres 4511    Fn wfn 5088   ` cfv 5093   reccrdg 6234  freccfrec 6255
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170  df-irdg 6235  df-frec 6256
This theorem is referenced by: (None)
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