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

Theorem tfrlem11 6420
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 4474 . 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 6419 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
5 fnfun 5357 . . . . . . . 8  |-  ( C  Fn  suc  dom recs ( F )  ->  Fun  C )
64, 5syl 15 . . . . . . 7  |-  ( dom recs
( F )  e.  On  ->  Fun  C )
7 ssun1 3351 . . . . . . . . 9  |- recs ( F )  C_  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )
87, 3sseqtr4i 3224 . . . . . . . 8  |- recs ( F )  C_  C
92tfrlem9 6417 . . . . . . . . 9  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
10 funssfv 5559 . . . . . . . . . . . 12  |-  ( ( Fun  C  /\ recs ( F )  C_  C  /\  B  e.  dom recs ( F ) )  -> 
( C `  B
)  =  (recs ( F ) `  B
) )
11103expa 1151 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  e.  dom recs ( F ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
1211adantrl 696 . . . . . . . . . 10  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
13 onelss 4450 . . . . . . . . . . . 12  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  B  C_  dom recs ( F ) ) )
1413imp 418 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) )  ->  B  C_  dom recs ( F ) )
15 fun2ssres 5311 . . . . . . . . . . . . 13  |-  ( ( Fun  C  /\ recs ( F )  C_  C  /\  B  C_  dom recs ( F ) )  -> 
( C  |`  B )  =  (recs ( F )  |`  B )
)
16153expa 1151 . . . . . . . . . . . 12  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( C  |`  B )  =  (recs ( F )  |`  B )
)
1716fveq2d 5545 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( F `  ( C  |`  B ) )  =  ( F `  (recs ( F )  |`  B ) ) )
1814, 17sylan2 460 . . . . . . . . . 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 2310 . . . . . . . . 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 212 . . . . . . . 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 662 . . . . . . 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 457 . . . . . 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 588 . . . . 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 43 . . . 4  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( B  e. 
dom recs ( F )  -> 
( C `  B
)  =  ( F `
 ( C  |`  B ) ) ) ) )
2524pm2.43d 44 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
26 opex 4253 . . . . . . . . 9  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  _V
2726snid 3680 . . . . . . . 8  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. B , 
( F `  ( C  |`  B ) )
>. }
28 opeq1 3812 . . . . . . . . . . 11  |-  ( B  =  dom recs ( F
)  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
2928adantl 452 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
30 eqimss 3243 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  B  C_  dom recs ( F ) )
318, 15mp3an2 1265 . . . . . . . . . . . . . 14  |-  ( ( Fun  C  /\  B  C_ 
dom recs ( F ) )  ->  ( C  |`  B )  =  (recs ( F )  |`  B ) )
326, 30, 31syl2an 463 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  =  (recs ( F )  |`  B ) )
33 reseq2 4966 . . . . . . . . . . . . . . 15  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  =  (recs ( F )  |`  dom recs ( F ) ) )
342tfrlem6 6414 . . . . . . . . . . . . . . . 16  |-  Rel recs ( F )
35 resdm 5009 . . . . . . . . . . . . . . . 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 2344 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  = recs ( F ) )
3837adantl 452 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  (recs ( F )  |`  B )  = recs ( F ) )
3932, 38eqtrd 2328 . . . . . . . . . . . 12  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  = recs ( F ) )
4039fveq2d 5545 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( F `  ( C  |`  B ) )  =  ( F `
recs ( F ) ) )
4140opeq2d 3819 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.  =  <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. )
4229, 41eqtrd 2328 . . . . . . . . 9  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. )
4342sneqd 3666 . . . . . . . 8  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  { <. B , 
( F `  ( C  |`  B ) )
>. }  =  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )
4427, 43syl5eleq 2382 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
45 elun2 3356 . . . . . . 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 15 . . . . . 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 2387 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  C )
484adantr 451 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  C  Fn  suc  dom recs ( F ) )
49 simpr 447 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  =  dom recs ( F ) )
50 sucidg 4486 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5150adantr 451 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5249, 51eqeltrd 2370 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  e.  suc  dom recs ( F ) )
53 fnopfvb 5580 . . . . . 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 642 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( ( C `  B )  =  ( F `  ( C  |`  B ) )  <->  <. B ,  ( F `  ( C  |`  B ) ) >.  e.  C ) )
5547, 54mpbird 223 . . . 4  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) )
5655ex 423 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  =  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
5725, 56jaod 369 . 2  |-  ( dom recs
( F )  e.  On  ->  ( ( B  e.  dom recs ( F )  \/  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
581, 57syl5 28 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    u. cun 3163    C_ wss 3165   {csn 3653   <.cop 3656   Oncon0 4408   suc csuc 4410   dom cdm 4705    |` cres 4707   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   ` cfv 5271  recscrecs 6403
This theorem is referenced by:  tfrlem12  6421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-recs 6404
  Copyright terms: Public domain W3C validator