MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem11 Unicode version

Theorem tfrlem11 6608
Description: Lemma for transfinite recursion. Compute the value of  C. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlem.3  |-  C  =  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
Assertion
Ref Expression
tfrlem11  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  suc  dom recs ( F
)  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
Distinct variable groups:    x, f,
y, B    C, f, x, y    f, F, x, y
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem11
StepHypRef Expression
1 elsuci 4607 . 2  |-  ( B  e.  suc  dom recs ( F )  ->  ( B  e.  dom recs ( F )  \/  B  =  dom recs ( F ) ) )
2 tfrlem.1 . . . . . . . . 9  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
3 tfrlem.3 . . . . . . . . 9  |-  C  =  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
42, 3tfrlem10 6607 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
5 fnfun 5501 . . . . . . . 8  |-  ( C  Fn  suc  dom recs ( F )  ->  Fun  C )
64, 5syl 16 . . . . . . 7  |-  ( dom recs
( F )  e.  On  ->  Fun  C )
7 ssun1 3470 . . . . . . . . 9  |- recs ( F )  C_  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )
87, 3sseqtr4i 3341 . . . . . . . 8  |- recs ( F )  C_  C
92tfrlem9 6605 . . . . . . . . 9  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
10 funssfv 5705 . . . . . . . . . . . 12  |-  ( ( Fun  C  /\ recs ( F )  C_  C  /\  B  e.  dom recs ( F ) )  -> 
( C `  B
)  =  (recs ( F ) `  B
) )
11103expa 1153 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  e.  dom recs ( F ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
1211adantrl 697 . . . . . . . . . 10  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
13 onelss 4583 . . . . . . . . . . . 12  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  B  C_  dom recs ( F ) ) )
1413imp 419 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) )  ->  B  C_  dom recs ( F ) )
15 fun2ssres 5453 . . . . . . . . . . . . 13  |-  ( ( Fun  C  /\ recs ( F )  C_  C  /\  B  C_  dom recs ( F ) )  -> 
( C  |`  B )  =  (recs ( F )  |`  B )
)
16153expa 1153 . . . . . . . . . . . 12  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( C  |`  B )  =  (recs ( F )  |`  B )
)
1716fveq2d 5691 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( F `  ( C  |`  B ) )  =  ( F `  (recs ( F )  |`  B ) ) )
1814, 17sylan2 461 . . . . . . . . . 10  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) ) )  ->  ( F `  ( C  |`  B ) )  =  ( F `
 (recs ( F )  |`  B )
) )
1912, 18eqeq12d 2418 . . . . . . . . 9  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) ) )  ->  ( ( C `
 B )  =  ( F `  ( C  |`  B ) )  <-> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) )
209, 19syl5ibr 213 . . . . . . . 8  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) ) )  ->  ( B  e. 
dom recs ( F )  -> 
( C `  B
)  =  ( F `
 ( C  |`  B ) ) ) )
218, 20mpanl2 663 . . . . . . 7  |-  ( ( Fun  C  /\  ( dom recs ( F )  e.  On  /\  B  e. 
dom recs ( F ) ) )  ->  ( B  e.  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
226, 21sylan 458 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  ( dom recs ( F )  e.  On  /\  B  e. 
dom recs ( F ) ) )  ->  ( B  e.  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
2322exp32 589 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  ( dom recs ( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( B  e.  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) ) ) )
2423pm2.43i 45 . . . 4  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( B  e. 
dom recs ( F )  -> 
( C `  B
)  =  ( F `
 ( C  |`  B ) ) ) ) )
2524pm2.43d 46 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
26 opex 4387 . . . . . . . . 9  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  _V
2726snid 3801 . . . . . . . 8  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. B , 
( F `  ( C  |`  B ) )
>. }
28 opeq1 3944 . . . . . . . . . . 11  |-  ( B  =  dom recs ( F
)  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
2928adantl 453 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
30 eqimss 3360 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  B  C_  dom recs ( F ) )
318, 15mp3an2 1267 . . . . . . . . . . . . . 14  |-  ( ( Fun  C  /\  B  C_ 
dom recs ( F ) )  ->  ( C  |`  B )  =  (recs ( F )  |`  B ) )
326, 30, 31syl2an 464 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  =  (recs ( F )  |`  B ) )
33 reseq2 5100 . . . . . . . . . . . . . . 15  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  =  (recs ( F )  |`  dom recs ( F ) ) )
342tfrlem6 6602 . . . . . . . . . . . . . . . 16  |-  Rel recs ( F )
35 resdm 5143 . . . . . . . . . . . . . . . 16  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
3634, 35ax-mp 8 . . . . . . . . . . . . . . 15  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
3733, 36syl6eq 2452 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  = recs ( F ) )
3837adantl 453 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  (recs ( F )  |`  B )  = recs ( F ) )
3932, 38eqtrd 2436 . . . . . . . . . . . 12  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  = recs ( F ) )
4039fveq2d 5691 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( F `  ( C  |`  B ) )  =  ( F `
recs ( F ) ) )
4140opeq2d 3951 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.  =  <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. )
4229, 41eqtrd 2436 . . . . . . . . 9  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. )
4342sneqd 3787 . . . . . . . 8  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  { <. B , 
( F `  ( C  |`  B ) )
>. }  =  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )
4427, 43syl5eleq 2490 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
45 elun2 3475 . . . . . . 7  |-  ( <. B ,  ( F `  ( C  |`  B ) ) >.  e.  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. }  ->  <. B ,  ( F `  ( C  |`  B ) ) >.  e.  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } ) )
4644, 45syl 16 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  (recs ( F )  u.  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } ) )
4746, 3syl6eleqr 2495 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  C )
484adantr 452 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  C  Fn  suc  dom recs ( F ) )
49 simpr 448 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  =  dom recs ( F ) )
50 sucidg 4619 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5150adantr 452 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5249, 51eqeltrd 2478 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  e.  suc  dom recs ( F ) )
53 fnopfvb 5727 . . . . . 6  |-  ( ( C  Fn  suc  dom recs ( F )  /\  B  e.  suc  dom recs ( F
) )  ->  (
( C `  B
)  =  ( F `
 ( C  |`  B ) )  <->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  C ) )
5448, 52, 53syl2anc 643 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( ( C `  B )  =  ( F `  ( C  |`  B ) )  <->  <. B ,  ( F `  ( C  |`  B ) ) >.  e.  C ) )
5547, 54mpbird 224 . . . 4  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) )
5655ex 424 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  =  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
5725, 56jaod 370 . 2  |-  ( dom recs
( F )  e.  On  ->  ( ( B  e.  dom recs ( F )  \/  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
581, 57syl5 30 1  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  suc  dom recs ( F
)  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667    u. cun 3278    C_ wss 3280   {csn 3774   <.cop 3777   Oncon0 4541   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:  tfrlem12  6609
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-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-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-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-fv 5421  df-recs 6592
  Copyright terms: Public domain W3C validator