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Theorem tfrlem11 7106
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 5489 . 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 7105 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
5 fnfun 5673 . . . . . . . 8  |-  ( C  Fn  suc  dom recs ( F )  ->  Fun  C )
64, 5syl 17 . . . . . . 7  |-  ( dom recs
( F )  e.  On  ->  Fun  C )
7 ssun1 3597 . . . . . . . . 9  |- recs ( F )  C_  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )
87, 3sseqtr4i 3465 . . . . . . . 8  |- recs ( F )  C_  C
92tfrlem9 7103 . . . . . . . . 9  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
10 funssfv 5880 . . . . . . . . . . . 12  |-  ( ( Fun  C  /\ recs ( F )  C_  C  /\  B  e.  dom recs ( F ) )  -> 
( C `  B
)  =  (recs ( F ) `  B
) )
11103expa 1208 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  e.  dom recs ( F ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
1211adantrl 722 . . . . . . . . . 10  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
13 onelss 5465 . . . . . . . . . . . 12  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  B  C_  dom recs ( F ) ) )
1413imp 431 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) )  ->  B  C_  dom recs ( F ) )
15 fun2ssres 5623 . . . . . . . . . . . . 13  |-  ( ( Fun  C  /\ recs ( F )  C_  C  /\  B  C_  dom recs ( F ) )  -> 
( C  |`  B )  =  (recs ( F )  |`  B )
)
16153expa 1208 . . . . . . . . . . . 12  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( C  |`  B )  =  (recs ( F )  |`  B )
)
1716fveq2d 5869 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( F `  ( C  |`  B ) )  =  ( F `  (recs ( F )  |`  B ) ) )
1814, 17sylan2 477 . . . . . . . . . 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 2466 . . . . . . . . 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 225 . . . . . . . 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 687 . . . . . . 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 474 . . . . . 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 610 . . . . 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 49 . . . 4  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( B  e. 
dom recs ( F )  -> 
( C `  B
)  =  ( F `
 ( C  |`  B ) ) ) ) )
2524pm2.43d 50 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
26 opex 4664 . . . . . . . . 9  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  _V
2726snid 3996 . . . . . . . 8  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. B , 
( F `  ( C  |`  B ) )
>. }
28 opeq1 4166 . . . . . . . . . . 11  |-  ( B  =  dom recs ( F
)  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
2928adantl 468 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
30 eqimss 3484 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  B  C_  dom recs ( F ) )
318, 15mp3an2 1352 . . . . . . . . . . . . . 14  |-  ( ( Fun  C  /\  B  C_ 
dom recs ( F ) )  ->  ( C  |`  B )  =  (recs ( F )  |`  B ) )
326, 30, 31syl2an 480 . . . . . . . . . . . . 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 7100 . . . . . . . . . . . . . . . 16  |-  Rel recs ( F )
35 resdm 5146 . . . . . . . . . . . . . . . 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 2501 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  = recs ( F ) )
3837adantl 468 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  (recs ( F )  |`  B )  = recs ( F ) )
3932, 38eqtrd 2485 . . . . . . . . . . . 12  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  = recs ( F ) )
4039fveq2d 5869 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( F `  ( C  |`  B ) )  =  ( F `
recs ( F ) ) )
4140opeq2d 4173 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.  =  <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. )
4229, 41eqtrd 2485 . . . . . . . . 9  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. )
4342sneqd 3980 . . . . . . . 8  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  { <. B , 
( F `  ( C  |`  B ) )
>. }  =  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )
4427, 43syl5eleq 2535 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
45 elun2 3602 . . . . . . 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 17 . . . . . 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 2540 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  C )
484adantr 467 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  C  Fn  suc  dom recs ( F ) )
49 simpr 463 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  =  dom recs ( F ) )
50 sucidg 5501 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5150adantr 467 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5249, 51eqeltrd 2529 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  e.  suc  dom recs ( F ) )
53 fnopfvb 5906 . . . . . 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 667 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( ( C `  B )  =  ( F `  ( C  |`  B ) )  <->  <. B ,  ( F `  ( C  |`  B ) ) >.  e.  C ) )
5547, 54mpbird 236 . . . 4  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) )
5655ex 436 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  =  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
5725, 56jaod 382 . 2  |-  ( dom recs
( F )  e.  On  ->  ( ( B  e.  dom recs ( F )  \/  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
581, 57syl5 33 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 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887   {cab 2437   A.wral 2737   E.wrex 2738    u. cun 3402    C_ wss 3404   {csn 3968   <.cop 3974   dom cdm 4834    |` cres 4836   Rel wrel 4839   Oncon0 5423   suc csuc 5425   Fun wfun 5576    Fn wfn 5577   ` cfv 5582  recscrecs 7089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-wrecs 7028  df-recs 7090
This theorem is referenced by:  tfrlem12  7107
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