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

Theorem tfrlem11 6958
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 4894 . 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 6957 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
5 fnfun 5617 . . . . . . . 8  |-  ( C  Fn  suc  dom recs ( F )  ->  Fun  C )
64, 5syl 16 . . . . . . 7  |-  ( dom recs
( F )  e.  On  ->  Fun  C )
7 ssun1 3628 . . . . . . . . 9  |- recs ( F )  C_  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )
87, 3sseqtr4i 3498 . . . . . . . 8  |- recs ( F )  C_  C
92tfrlem9 6955 . . . . . . . . 9  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
10 funssfv 5815 . . . . . . . . . . . 12  |-  ( ( Fun  C  /\ recs ( F )  C_  C  /\  B  e.  dom recs ( F ) )  -> 
( C `  B
)  =  (recs ( F ) `  B
) )
11103expa 1188 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  e.  dom recs ( F ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
1211adantrl 715 . . . . . . . . . 10  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
13 onelss 4870 . . . . . . . . . . . 12  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  B  C_  dom recs ( F ) ) )
1413imp 429 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) )  ->  B  C_  dom recs ( F ) )
15 fun2ssres 5568 . . . . . . . . . . . . 13  |-  ( ( Fun  C  /\ recs ( F )  C_  C  /\  B  C_  dom recs ( F ) )  -> 
( C  |`  B )  =  (recs ( F )  |`  B )
)
16153expa 1188 . . . . . . . . . . . 12  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( C  |`  B )  =  (recs ( F )  |`  B )
)
1716fveq2d 5804 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( F `  ( C  |`  B ) )  =  ( F `  (recs ( F )  |`  B ) ) )
1814, 17sylan2 474 . . . . . . . . . 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 2476 . . . . . . . . 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 221 . . . . . . . 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 681 . . . . . . 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 471 . . . . . 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 605 . . . . 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 47 . . . 4  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( B  e. 
dom recs ( F )  -> 
( C `  B
)  =  ( F `
 ( C  |`  B ) ) ) ) )
2524pm2.43d 48 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
26 opex 4665 . . . . . . . . 9  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  _V
2726snid 4014 . . . . . . . 8  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. B , 
( F `  ( C  |`  B ) )
>. }
28 opeq1 4168 . . . . . . . . . . 11  |-  ( B  =  dom recs ( F
)  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
2928adantl 466 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
30 eqimss 3517 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  B  C_  dom recs ( F ) )
318, 15mp3an2 1303 . . . . . . . . . . . . . 14  |-  ( ( Fun  C  /\  B  C_ 
dom recs ( F ) )  ->  ( C  |`  B )  =  (recs ( F )  |`  B ) )
326, 30, 31syl2an 477 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  =  (recs ( F )  |`  B ) )
33 reseq2 5214 . . . . . . . . . . . . . . 15  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  =  (recs ( F )  |`  dom recs ( F ) ) )
342tfrlem6 6952 . . . . . . . . . . . . . . . 16  |-  Rel recs ( F )
35 resdm 5257 . . . . . . . . . . . . . . . 16  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
3634, 35ax-mp 5 . . . . . . . . . . . . . . 15  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
3733, 36syl6eq 2511 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  = recs ( F ) )
3837adantl 466 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  (recs ( F )  |`  B )  = recs ( F ) )
3932, 38eqtrd 2495 . . . . . . . . . . . 12  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  = recs ( F ) )
4039fveq2d 5804 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( F `  ( C  |`  B ) )  =  ( F `
recs ( F ) ) )
4140opeq2d 4175 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.  =  <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. )
4229, 41eqtrd 2495 . . . . . . . . 9  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. )
4342sneqd 3998 . . . . . . . 8  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  { <. B , 
( F `  ( C  |`  B ) )
>. }  =  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )
4427, 43syl5eleq 2548 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
45 elun2 3633 . . . . . . 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 2553 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  C )
484adantr 465 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  C  Fn  suc  dom recs ( F ) )
49 simpr 461 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  =  dom recs ( F ) )
50 sucidg 4906 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5150adantr 465 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5249, 51eqeltrd 2542 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  e.  suc  dom recs ( F ) )
53 fnopfvb 5843 . . . . . 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 661 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( ( C `  B )  =  ( F `  ( C  |`  B ) )  <->  <. B ,  ( F `  ( C  |`  B ) ) >.  e.  C ) )
5547, 54mpbird 232 . . . 4  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) )
5655ex 434 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  =  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
5725, 56jaod 380 . 2  |-  ( dom recs
( F )  e.  On  ->  ( ( B  e.  dom recs ( F )  \/  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
581, 57syl5 32 1  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  suc  dom recs ( F
)  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800    u. cun 3435    C_ wss 3437   {csn 3986   <.cop 3992   Oncon0 4828   suc csuc 4830   dom cdm 4949    |` cres 4951   Rel wrel 4954   Fun wfun 5521    Fn wfn 5522   ` cfv 5527  recscrecs 6942
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535  df-recs 6943
This theorem is referenced by:  tfrlem12  6959
  Copyright terms: Public domain W3C validator