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Theorem tfrlem16 6613
Description: Lemma for finite recursion. Without assuming ax-rep 4280, 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 6604 . . 3  |-  Ord  dom recs ( F )
3 ordzsl 4784 . . 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 5109 . . . . . . 7  |-  (recs ( F )  |`  (/) )  =  (/)
6 0ex 4299 . . . . . . 7  |-  (/)  e.  _V
75, 6eqeltri 2474 . . . . . 6  |-  (recs ( F )  |`  (/) )  e. 
_V
8 0elon 4594 . . . . . . 7  |-  (/)  e.  On
91tfrlem15 6612 . . . . . . 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 3593 . . . . 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 6610 . . . . 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 4547 . . . . . . . . . 10  |-  suc  z  =  ( z  u. 
{ z } )
1816, 17syl6eq 2452 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  ->  dom recs ( F )  =  ( z  u.  {
z } ) )
1918reseq2d 5105 . . . . . . . 8  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
(recs ( F )  |`  dom recs ( F ) )  =  (recs ( F )  |`  (
z  u.  { z } ) ) )
201tfrlem6 6602 . . . . . . . . 9  |-  Rel recs ( F )
21 resdm 5143 . . . . . . . . 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 5119 . . . . . . . 8  |-  (recs ( F )  |`  (
z  u.  { z } ) )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )
2419, 22, 233eqtr3g 2459 . . . . . . 7  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) ) )
25 vex 2919 . . . . . . . . . . 11  |-  z  e. 
_V
2625sucid 4620 . . . . . . . . . 10  |-  z  e. 
suc  z
2726, 16syl5eleqr 2491 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
z  e.  dom recs ( F ) )
281tfrlem9a 6606 . . . . . . . . 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 4365 . . . . . . . . 9  |-  { <. z ,  (recs ( F ) `  z )
>. }  e.  _V
311tfrlem7 6603 . . . . . . . . . 10  |-  Fun recs ( F )
32 funressn 5878 . . . . . . . . . 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 4308 . . . . . . . 8  |-  (recs ( F )  |`  { z } )  e.  _V
35 unexg 4669 . . . . . . . 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 2478 . . . . . 6  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  e.  _V )
3837rexlimiva 2785 . . . . 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 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916    u. cun 3278    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   Ord word 4540   Oncon0 4541   Lim wlim 4542   suc csuc 4543   dom cdm 4837    |` cres 4839   Rel wrel 4842   Fun wfun 5407    Fn wfn 5408   ` cfv 5413  recscrecs 6591
This theorem is referenced by:  tfr1a  6614
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592
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