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Theorem tfrlem11 6399
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 4457 . 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 6398 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
5 fnfun 5306 . . . . . . . 8  |-  ( C  Fn  suc  dom recs ( F )  ->  Fun  C )
64, 5syl 17 . . . . . . 7  |-  ( dom recs
( F )  e.  On  ->  Fun  C )
7 ssun1 3339 . . . . . . . . 9  |- recs ( F )  C_  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )
87, 3sseqtr4i 3212 . . . . . . . 8  |- recs ( F )  C_  C
92tfrlem9 6396 . . . . . . . . 9  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
10 funssfv 5503 . . . . . . . . . . . 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 698 . . . . . . . . . 10  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) ) )  ->  ( C `  B )  =  (recs ( F ) `  B ) )
13 onelss 4433 . . . . . . . . . . . 12  |-  ( dom recs
( F )  e.  On  ->  ( B  e.  dom recs ( F )  ->  B  C_  dom recs ( F ) ) )
1413imp 420 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  e.  dom recs ( F ) )  ->  B  C_  dom recs ( F ) )
15 fun2ssres 5260 . . . . . . . . . . . . 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 5489 . . . . . . . . . . 11  |-  ( ( ( Fun  C  /\ recs ( F )  C_  C
)  /\  B  C_  dom recs ( F ) )  -> 
( F `  ( C  |`  B ) )  =  ( F `  (recs ( F )  |`  B ) ) )
1814, 17sylan2 462 . . . . . . . . . 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 2298 . . . . . . . . 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 214 . . . . . . . 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 664 . . . . . . 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 459 . . . . . 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 590 . . . . 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 4236 . . . . . . . . 9  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  _V
2726snid 3668 . . . . . . . 8  |-  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. B , 
( F `  ( C  |`  B ) )
>. }
28 opeq1 3797 . . . . . . . . . . 11  |-  ( B  =  dom recs ( F
)  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
2928adantl 454 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.
)
30 eqimss 3231 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  =  (recs ( F )  |`  B ) )
33 reseq2 4949 . . . . . . . . . . . . . . 15  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  =  (recs ( F )  |`  dom recs ( F ) ) )
342tfrlem6 6393 . . . . . . . . . . . . . . . 16  |-  Rel recs ( F )
35 resdm 4992 . . . . . . . . . . . . . . . 16  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
3634, 35ax-mp 10 . . . . . . . . . . . . . . 15  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
3733, 36syl6eq 2332 . . . . . . . . . . . . . 14  |-  ( B  =  dom recs ( F
)  ->  (recs ( F )  |`  B )  = recs ( F ) )
3837adantl 454 . . . . . . . . . . . . 13  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  (recs ( F )  |`  B )  = recs ( F ) )
3932, 38eqtrd 2316 . . . . . . . . . . . 12  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C  |`  B )  = recs ( F ) )
4039fveq2d 5489 . . . . . . . . . . 11  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( F `  ( C  |`  B ) )  =  ( F `
recs ( F ) ) )
4140opeq2d 3804 . . . . . . . . . 10  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. dom recs ( F ) ,  ( F `  ( C  |`  B ) ) >.  =  <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. )
4229, 41eqtrd 2316 . . . . . . . . 9  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  =  <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. )
4342sneqd 3654 . . . . . . . 8  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  { <. B , 
( F `  ( C  |`  B ) )
>. }  =  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )
4427, 43syl5eleq 2370 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
45 elun2 3344 . . . . . . 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 2375 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  <. B , 
( F `  ( C  |`  B ) )
>.  e.  C )
484adantr 453 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  C  Fn  suc  dom recs ( F ) )
49 simpr 449 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  =  dom recs ( F ) )
50 sucidg 4469 . . . . . . . 8  |-  ( dom recs
( F )  e.  On  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5150adantr 453 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
5249, 51eqeltrd 2358 . . . . . 6  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  B  e.  suc  dom recs ( F ) )
53 fnopfvb 5525 . . . . . 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 644 . . . . 5  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( ( C `  B )  =  ( F `  ( C  |`  B ) )  <->  <. B ,  ( F `  ( C  |`  B ) ) >.  e.  C ) )
5547, 54mpbird 225 . . . 4  |-  ( ( dom recs ( F )  e.  On  /\  B  =  dom recs ( F ) )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) )
5655ex 425 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( B  =  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
5725, 56jaod 371 . 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 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   {cab 2270   A.wral 2544   E.wrex 2545    u. cun 3151    C_ wss 3153   {csn 3641   <.cop 3644   Oncon0 4391   suc csuc 4393   dom cdm 4688    |` cres 4690   Rel wrel 4693   Fun wfun 5215    Fn wfn 5216   ` cfv 5221  recscrecs 6382
This theorem is referenced by:  tfrlem12  6400
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229  df-recs 6383
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