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Theorem tfrlem16 7059
Description: Lemma for finite recursion. Without assuming ax-rep 4558, 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 7050 . . 3  |-  Ord  dom recs ( F )
3 ordzsl 6658 . . 3  |-  ( Ord 
dom recs ( F )  <->  ( dom recs ( F )  =  (/)  \/ 
E. z  e.  On  dom recs ( F )  =  suc  z  \/  Lim  dom recs
( F ) ) )
42, 3mpbi 208 . 2  |-  ( dom recs
( F )  =  (/)  \/  E. z  e.  On  dom recs ( F
)  =  suc  z  \/  Lim  dom recs ( F
) )
5 res0 5276 . . . . . . 7  |-  (recs ( F )  |`  (/) )  =  (/)
6 0ex 4577 . . . . . . 7  |-  (/)  e.  _V
75, 6eqeltri 2551 . . . . . 6  |-  (recs ( F )  |`  (/) )  e. 
_V
8 0elon 4931 . . . . . . 7  |-  (/)  e.  On
91tfrlem15 7058 . . . . . . 7  |-  ( (/)  e.  On  ->  ( (/)  e.  dom recs ( F )  <->  (recs ( F )  |`  (/) )  e. 
_V ) )
108, 9ax-mp 5 . . . . . 6  |-  ( (/)  e.  dom recs ( F )  <-> 
(recs ( F )  |`  (/) )  e.  _V )
117, 10mpbir 209 . . . . 5  |-  (/)  e.  dom recs ( F )
12 n0i 3790 . . . . 5  |-  ( (/)  e.  dom recs ( F )  ->  -.  dom recs ( F )  =  (/) )
1311, 12ax-mp 5 . . . 4  |-  -.  dom recs ( F )  =  (/)
1413pm2.21i 131 . . 3  |-  ( dom recs
( F )  =  (/)  ->  Lim  dom recs ( F ) )
151tfrlem13 7056 . . . . 5  |-  -. recs ( F )  e.  _V
16 simpr 461 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  ->  dom recs ( F )  =  suc  z )
17 df-suc 4884 . . . . . . . . . 10  |-  suc  z  =  ( z  u. 
{ z } )
1816, 17syl6eq 2524 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  ->  dom recs ( F )  =  ( z  u.  {
z } ) )
1918reseq2d 5271 . . . . . . . 8  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
(recs ( F )  |`  dom recs ( F ) )  =  (recs ( F )  |`  (
z  u.  { z } ) ) )
201tfrlem6 7048 . . . . . . . . 9  |-  Rel recs ( F )
21 resdm 5313 . . . . . . . . 9  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
2220, 21ax-mp 5 . . . . . . . 8  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
23 resundi 5285 . . . . . . . 8  |-  (recs ( F )  |`  (
z  u.  { z } ) )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )
2419, 22, 233eqtr3g 2531 . . . . . . 7  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) ) )
25 vex 3116 . . . . . . . . . . 11  |-  z  e. 
_V
2625sucid 4957 . . . . . . . . . 10  |-  z  e. 
suc  z
2726, 16syl5eleqr 2562 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
z  e.  dom recs ( F ) )
281tfrlem9a 7052 . . . . . . . . 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 4688 . . . . . . . . 9  |-  { <. z ,  (recs ( F ) `  z )
>. }  e.  _V
311tfrlem7 7049 . . . . . . . . . 10  |-  Fun recs ( F )
32 funressn 6072 . . . . . . . . . 10  |-  ( Fun recs
( F )  -> 
(recs ( F )  |`  { z } ) 
C_  { <. z ,  (recs ( F ) `
 z ) >. } )
3331, 32ax-mp 5 . . . . . . . . 9  |-  (recs ( F )  |`  { z } )  C_  { <. z ,  (recs ( F ) `  z )
>. }
3430, 33ssexi 4592 . . . . . . . 8  |-  (recs ( F )  |`  { z } )  e.  _V
35 unexg 6583 . . . . . . . 8  |-  ( ( (recs ( F )  |`  z )  e.  _V  /\  (recs ( F )  |`  { z } )  e.  _V )  -> 
( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )  e.  _V )
3629, 34, 35sylancl 662 . . . . . . 7  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )  e.  _V )
3724, 36eqeltrd 2555 . . . . . 6  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  e.  _V )
3837rexlimiva 2951 . . . . 5  |-  ( E. z  e.  On  dom recs ( F )  =  suc  z  -> recs ( F )  e.  _V )
3915, 38mto 176 . . . 4  |-  -.  E. z  e.  On  dom recs ( F )  =  suc  z
4039pm2.21i 131 . . 3  |-  ( E. z  e.  On  dom recs ( F )  =  suc  z  ->  Lim  dom recs ( F ) )
41 id 22 . . 3  |-  ( Lim 
dom recs ( F )  ->  Lim  dom recs ( F ) )
4214, 40, 413jaoi 1291 . 2  |-  ( ( dom recs ( F )  =  (/)  \/  E. z  e.  On  dom recs ( F
)  =  suc  z  \/  Lim  dom recs ( F
) )  ->  Lim  dom recs
( F ) )
434, 42ax-mp 5 1  |-  Lim  dom recs ( F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   Ord word 4877   Oncon0 4878   Lim wlim 4879   suc csuc 4880   dom cdm 4999    |` cres 5001   Rel wrel 5004   Fun wfun 5580    Fn wfn 5581   ` cfv 5586  recscrecs 7038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-recs 7039
This theorem is referenced by:  tfr1a  7060
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