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Theorem tfrlem16 6590
Description: Lemma for finite recursion. Without assuming ax-rep 4261, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem16  |-  Lim  dom recs ( F )
Distinct variable group:    x, f, y, F
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem16
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem8 6581 . . 3  |-  Ord  dom recs ( F )
3 ordzsl 4765 . . 3  |-  ( Ord 
dom recs ( F )  <->  ( dom recs ( F )  =  (/)  \/ 
E. z  e.  On  dom recs ( F )  =  suc  z  \/  Lim  dom recs
( F ) ) )
42, 3mpbi 200 . 2  |-  ( dom recs
( F )  =  (/)  \/  E. z  e.  On  dom recs ( F
)  =  suc  z  \/  Lim  dom recs ( F
) )
5 res0 5090 . . . . . . 7  |-  (recs ( F )  |`  (/) )  =  (/)
6 0ex 4280 . . . . . . 7  |-  (/)  e.  _V
75, 6eqeltri 2457 . . . . . 6  |-  (recs ( F )  |`  (/) )  e. 
_V
8 0elon 4575 . . . . . . 7  |-  (/)  e.  On
91tfrlem15 6589 . . . . . . 7  |-  ( (/)  e.  On  ->  ( (/)  e.  dom recs ( F )  <->  (recs ( F )  |`  (/) )  e. 
_V ) )
108, 9ax-mp 8 . . . . . 6  |-  ( (/)  e.  dom recs ( F )  <-> 
(recs ( F )  |`  (/) )  e.  _V )
117, 10mpbir 201 . . . . 5  |-  (/)  e.  dom recs ( F )
12 n0i 3576 . . . . 5  |-  ( (/)  e.  dom recs ( F )  ->  -.  dom recs ( F )  =  (/) )
1311, 12ax-mp 8 . . . 4  |-  -.  dom recs ( F )  =  (/)
1413pm2.21i 125 . . 3  |-  ( dom recs
( F )  =  (/)  ->  Lim  dom recs ( F ) )
151tfrlem13 6587 . . . . 5  |-  -. recs ( F )  e.  _V
16 simpr 448 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  ->  dom recs ( F )  =  suc  z )
17 df-suc 4528 . . . . . . . . . 10  |-  suc  z  =  ( z  u. 
{ z } )
1816, 17syl6eq 2435 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  ->  dom recs ( F )  =  ( z  u.  {
z } ) )
1918reseq2d 5086 . . . . . . . 8  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
(recs ( F )  |`  dom recs ( F ) )  =  (recs ( F )  |`  (
z  u.  { z } ) ) )
201tfrlem6 6579 . . . . . . . . 9  |-  Rel recs ( F )
21 resdm 5124 . . . . . . . . 9  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
2220, 21ax-mp 8 . . . . . . . 8  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
23 resundi 5100 . . . . . . . 8  |-  (recs ( F )  |`  (
z  u.  { z } ) )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )
2419, 22, 233eqtr3g 2442 . . . . . . 7  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) ) )
25 vex 2902 . . . . . . . . . . 11  |-  z  e. 
_V
2625sucid 4601 . . . . . . . . . 10  |-  z  e. 
suc  z
2726, 16syl5eleqr 2474 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
z  e.  dom recs ( F ) )
281tfrlem9a 6583 . . . . . . . . 9  |-  ( z  e.  dom recs ( F
)  ->  (recs ( F )  |`  z
)  e.  _V )
2927, 28syl 16 . . . . . . . 8  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
(recs ( F )  |`  z )  e.  _V )
30 snex 4346 . . . . . . . . 9  |-  { <. z ,  (recs ( F ) `  z )
>. }  e.  _V
311tfrlem7 6580 . . . . . . . . . 10  |-  Fun recs ( F )
32 funressn 5858 . . . . . . . . . 10  |-  ( Fun recs
( F )  -> 
(recs ( F )  |`  { z } ) 
C_  { <. z ,  (recs ( F ) `
 z ) >. } )
3331, 32ax-mp 8 . . . . . . . . 9  |-  (recs ( F )  |`  { z } )  C_  { <. z ,  (recs ( F ) `  z )
>. }
3430, 33ssexi 4289 . . . . . . . 8  |-  (recs ( F )  |`  { z } )  e.  _V
35 unexg 4650 . . . . . . . 8  |-  ( ( (recs ( F )  |`  z )  e.  _V  /\  (recs ( F )  |`  { z } )  e.  _V )  -> 
( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )  e.  _V )
3629, 34, 35sylancl 644 . . . . . . 7  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )  e.  _V )
3724, 36eqeltrd 2461 . . . . . 6  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  e.  _V )
3837rexlimiva 2768 . . . . 5  |-  ( E. z  e.  On  dom recs ( F )  =  suc  z  -> recs ( F )  e.  _V )
3915, 38mto 169 . . . 4  |-  -.  E. z  e.  On  dom recs ( F )  =  suc  z
4039pm2.21i 125 . . 3  |-  ( E. z  e.  On  dom recs ( F )  =  suc  z  ->  Lim  dom recs ( F ) )
41 id 20 . . 3  |-  ( Lim 
dom recs ( F )  ->  Lim  dom recs ( F ) )
4214, 40, 413jaoi 1247 . 2  |-  ( ( dom recs ( F )  =  (/)  \/  E. z  e.  On  dom recs ( F
)  =  suc  z  \/  Lim  dom recs ( F
) )  ->  Lim  dom recs
( F ) )
434, 42ax-mp 8 1  |-  Lim  dom recs ( F )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1717   {cab 2373   A.wral 2649   E.wrex 2650   _Vcvv 2899    u. cun 3261    C_ wss 3263   (/)c0 3571   {csn 3757   <.cop 3760   Ord word 4521   Oncon0 4522   Lim wlim 4523   suc csuc 4524   dom cdm 4818    |` cres 4820   Rel wrel 4823   Fun wfun 5388    Fn wfn 5389   ` cfv 5394  recscrecs 6568
This theorem is referenced by:  tfr1a  6591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-recs 6569
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