MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem16 Unicode version

Theorem tfrlem16 6363
Description: Lemma for finite recursion. Without assuming ax-rep 4091, 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
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 6354 . . 3  |-  Ord  dom recs ( F )
3 ordzsl 4594 . . 3  |-  ( Ord 
dom recs ( F )  <->  ( dom recs ( F )  =  (/)  \/ 
E. z  e.  On  dom recs ( F )  =  suc  z  \/  Lim  dom recs
( F ) ) )
42, 3mpbi 201 . 2  |-  ( dom recs
( F )  =  (/)  \/  E. z  e.  On  dom recs ( F
)  =  suc  z  \/  Lim  dom recs ( F
) )
5 res0 4933 . . . . . . 7  |-  (recs ( F )  |`  (/) )  =  (/)
6 0ex 4110 . . . . . . 7  |-  (/)  e.  _V
75, 6eqeltri 2326 . . . . . 6  |-  (recs ( F )  |`  (/) )  e. 
_V
8 0elon 4403 . . . . . . 7  |-  (/)  e.  On
91tfrlem15 6362 . . . . . . 7  |-  ( (/)  e.  On  ->  ( (/)  e.  dom recs ( F )  <->  (recs ( F )  |`  (/) )  e. 
_V ) )
108, 9ax-mp 10 . . . . . 6  |-  ( (/)  e.  dom recs ( F )  <-> 
(recs ( F )  |`  (/) )  e.  _V )
117, 10mpbir 202 . . . . 5  |-  (/)  e.  dom recs ( F )
12 n0i 3421 . . . . 5  |-  ( (/)  e.  dom recs ( F )  ->  -.  dom recs ( F )  =  (/) )
1311, 12ax-mp 10 . . . 4  |-  -.  dom recs ( F )  =  (/)
1413pm2.21i 125 . . 3  |-  ( dom recs
( F )  =  (/)  ->  Lim  dom recs ( F ) )
151tfrlem13 6360 . . . . 5  |-  -. recs ( F )  e.  _V
16 simpr 449 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  ->  dom recs ( F )  =  suc  z )
17 df-suc 4356 . . . . . . . . . 10  |-  suc  z  =  ( z  u. 
{ z } )
1816, 17syl6eq 2304 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  ->  dom recs ( F )  =  ( z  u.  {
z } ) )
1918reseq2d 4929 . . . . . . . 8  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
(recs ( F )  |`  dom recs ( F ) )  =  (recs ( F )  |`  (
z  u.  { z } ) ) )
201tfrlem6 6352 . . . . . . . . 9  |-  Rel recs ( F )
21 resdm 4967 . . . . . . . . 9  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
2220, 21ax-mp 10 . . . . . . . 8  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
23 resundi 4943 . . . . . . . 8  |-  (recs ( F )  |`  (
z  u.  { z } ) )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )
2419, 22, 233eqtr3g 2311 . . . . . . 7  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) ) )
25 vex 2760 . . . . . . . . . . 11  |-  z  e. 
_V
2625sucid 4429 . . . . . . . . . 10  |-  z  e. 
suc  z
2726, 16syl5eleqr 2343 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
z  e.  dom recs ( F ) )
281tfrlem9a 6356 . . . . . . . . 9  |-  ( z  e.  dom recs ( F
)  ->  (recs ( F )  |`  z
)  e.  _V )
2927, 28syl 17 . . . . . . . 8  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
(recs ( F )  |`  z )  e.  _V )
30 snex 4174 . . . . . . . . 9  |-  { <. z ,  (recs ( F ) `  z )
>. }  e.  _V
311tfrlem7 6353 . . . . . . . . . 10  |-  Fun recs ( F )
32 funressn 5626 . . . . . . . . . 10  |-  ( Fun recs
( F )  -> 
(recs ( F )  |`  { z } ) 
C_  { <. z ,  (recs ( F ) `
 z ) >. } )
3331, 32ax-mp 10 . . . . . . . . 9  |-  (recs ( F )  |`  { z } )  C_  { <. z ,  (recs ( F ) `  z )
>. }
3430, 33ssexi 4119 . . . . . . . 8  |-  (recs ( F )  |`  { z } )  e.  _V
35 unexg 4479 . . . . . . . 8  |-  ( ( (recs ( F )  |`  z )  e.  _V  /\  (recs ( F )  |`  { z } )  e.  _V )  -> 
( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )  e.  _V )
3629, 34, 35sylancl 646 . . . . . . 7  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )  e.  _V )
3724, 36eqeltrd 2330 . . . . . 6  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  e.  _V )
3837rexlimiva 2635 . . . . 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 21 . . 3  |-  ( Lim 
dom recs ( F )  ->  Lim  dom recs ( F ) )
4214, 40, 413jaoi 1250 . 2  |-  ( ( dom recs ( F )  =  (/)  \/  E. z  e.  On  dom recs ( F
)  =  suc  z  \/  Lim  dom recs ( F
) )  ->  Lim  dom recs
( F ) )
434, 42ax-mp 10 1  |-  Lim  dom recs ( F )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360    \/ w3o 938    = wceq 1619    e. wcel 1621   {cab 2242   A.wral 2516   E.wrex 2517   _Vcvv 2757    u. cun 3111    C_ wss 3113   (/)c0 3416   {csn 3600   <.cop 3603   Ord word 4349   Oncon0 4350   Lim wlim 4351   suc csuc 4352   dom cdm 4647    |` cres 4649   Rel wrel 4652   Fun wfun 4653    Fn wfn 4654   ` cfv 4659  recscrecs 6341
This theorem is referenced by:  tfr1a  6364
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-recs 6342
  Copyright terms: Public domain W3C validator