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Theorem tfrlem16 6955
Description: Lemma for finite recursion. Without assuming ax-rep 4504, 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 6946 . . 3  |-  Ord  dom recs ( F )
3 ordzsl 6559 . . 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 5216 . . . . . . 7  |-  (recs ( F )  |`  (/) )  =  (/)
6 0ex 4523 . . . . . . 7  |-  (/)  e.  _V
75, 6eqeltri 2535 . . . . . 6  |-  (recs ( F )  |`  (/) )  e. 
_V
8 0elon 4873 . . . . . . 7  |-  (/)  e.  On
91tfrlem15 6954 . . . . . . 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 3743 . . . . 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 6952 . . . . 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 4826 . . . . . . . . . 10  |-  suc  z  =  ( z  u. 
{ z } )
1816, 17syl6eq 2508 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  ->  dom recs ( F )  =  ( z  u.  {
z } ) )
1918reseq2d 5211 . . . . . . . 8  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
(recs ( F )  |`  dom recs ( F ) )  =  (recs ( F )  |`  (
z  u.  { z } ) ) )
201tfrlem6 6944 . . . . . . . . 9  |-  Rel recs ( F )
21 resdm 5249 . . . . . . . . 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 5225 . . . . . . . 8  |-  (recs ( F )  |`  (
z  u.  { z } ) )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) )
2419, 22, 233eqtr3g 2515 . . . . . . 7  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  =  ( (recs ( F )  |`  z )  u.  (recs ( F )  |`  { z } ) ) )
25 vex 3074 . . . . . . . . . . 11  |-  z  e. 
_V
2625sucid 4899 . . . . . . . . . 10  |-  z  e. 
suc  z
2726, 16syl5eleqr 2546 . . . . . . . . 9  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> 
z  e.  dom recs ( F ) )
281tfrlem9a 6948 . . . . . . . . 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 4634 . . . . . . . . 9  |-  { <. z ,  (recs ( F ) `  z )
>. }  e.  _V
311tfrlem7 6945 . . . . . . . . . 10  |-  Fun recs ( F )
32 funressn 5997 . . . . . . . . . 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 4538 . . . . . . . 8  |-  (recs ( F )  |`  { z } )  e.  _V
35 unexg 6484 . . . . . . . 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 2539 . . . . . 6  |-  ( ( z  e.  On  /\  dom recs ( F )  =  suc  z )  -> recs ( F )  e.  _V )
3837rexlimiva 2935 . . . . 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 1282 . 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 964    = wceq 1370    e. wcel 1758   {cab 2436   A.wral 2795   E.wrex 2796   _Vcvv 3071    u. cun 3427    C_ wss 3429   (/)c0 3738   {csn 3978   <.cop 3984   Ord word 4819   Oncon0 4820   Lim wlim 4821   suc csuc 4822   dom cdm 4941    |` cres 4943   Rel wrel 4946   Fun wfun 5513    Fn wfn 5514   ` cfv 5519  recscrecs 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-recs 6935
This theorem is referenced by:  tfr1a  6956
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