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Theorem tfrlem9 6401
Description: Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
Hypothesis
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem9  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
Distinct variable groups:    x, f,
y, B    f, F, x, y
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem9
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eldm2g 4875 . . 3  |-  ( B  e.  dom recs ( F
)  ->  ( B  e.  dom recs ( F )  <->  E. z <. B ,  z
>.  e. recs ( F ) ) )
21ibi 232 . 2  |-  ( B  e.  dom recs ( F
)  ->  E. z <. B ,  z >.  e. recs ( F ) )
3 df-recs 6388 . . . . . 6  |- recs ( F )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
43eleq2i 2347 . . . . 5  |-  ( <. B ,  z >.  e. recs
( F )  <->  <. B , 
z >.  e.  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) } )
5 eluniab 3839 . . . . 5  |-  ( <. B ,  z >.  e. 
U. { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) }  <->  E. f ( <. B ,  z >.  e.  f  /\  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) ) )
64, 5bitri 240 . . . 4  |-  ( <. B ,  z >.  e. recs
( F )  <->  E. f
( <. B ,  z
>.  e.  f  /\  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) ) )
7 fnop 5347 . . . . . . . . . . . . . 14  |-  ( ( f  Fn  x  /\  <. B ,  z >.  e.  f )  ->  B  e.  x )
8 rspe 2604 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  ->  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
9 tfrlem.1 . . . . . . . . . . . . . . . . . 18  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
109abeq2i 2390 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  <->  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
11 elssuni 3855 . . . . . . . . . . . . . . . . . 18  |-  ( f  e.  A  ->  f  C_ 
U. A )
129recsfval 6397 . . . . . . . . . . . . . . . . . 18  |- recs ( F )  =  U. A
1311, 12syl6sseqr 3225 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  ->  f  C_ recs
( F ) )
1410, 13sylbir 204 . . . . . . . . . . . . . . . 16  |-  ( E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) )  -> 
f  C_ recs ( F
) )
158, 14syl 15 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  ->  f  C_ recs ( F ) )
16 fveq2 5525 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  B  ->  (
f `  y )  =  ( f `  B ) )
17 reseq2 4950 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  B  ->  (
f  |`  y )  =  ( f  |`  B ) )
1817fveq2d 5529 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  B  ->  ( F `  ( f  |`  y ) )  =  ( F `  (
f  |`  B ) ) )
1916, 18eqeq12d 2297 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  B  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( f `  B )  =  ( F `  ( f  |`  B ) ) ) )
2019rspcv 2880 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  x  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  -> 
( f `  B
)  =  ( F `
 ( f  |`  B ) ) ) )
21 fndm 5343 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  Fn  x  ->  dom  f  =  x )
2221eleq2d 2350 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  Fn  x  ->  ( B  e.  dom  f  <->  B  e.  x ) )
239tfrlem7 6399 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  Fun recs ( F )
24 funssfv 5543 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  e. 
dom  f )  -> 
(recs ( F ) `
 B )  =  ( f `  B
) )
2523, 24mp3an1 1264 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f  C_ recs ( F
)  /\  B  e.  dom  f )  ->  (recs ( F ) `  B
)  =  ( f `
 B ) )
2625adantrl 696 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
(recs ( F ) `
 B )  =  ( f `  B
) )
2721eleq1d 2349 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( f  Fn  x  ->  ( dom  f  e.  On  <->  x  e.  On ) )
28 onelss 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( dom  f  e.  On  ->  ( B  e.  dom  f  ->  B  C_  dom  f ) )
2927, 28syl6bir 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( f  Fn  x  ->  (
x  e.  On  ->  ( B  e.  dom  f  ->  B  C_  dom  f ) ) )
3029imp31 421 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f )  ->  B  C_ 
dom  f )
31 fun2ssres 5295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  C_  dom  f )  ->  (recs ( F )  |`  B )  =  ( f  |`  B ) )
3231fveq2d 5529 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  C_  dom  f )  ->  ( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3323, 32mp3an1 1264 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f  C_ recs ( F
)  /\  B  C_  dom  f )  ->  ( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3430, 33sylan2 460 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3526, 34eqeq12d 2297 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
( (recs ( F ) `  B )  =  ( F `  (recs ( F )  |`  B ) )  <->  ( f `  B )  =  ( F `  ( f  |`  B ) ) ) )
3635exbiri 605 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f 
C_ recs ( F )  ->  ( ( ( f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f )  ->  (
( f `  B
)  =  ( F `
 ( f  |`  B ) )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) ) )
3736com3l 75 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f )  ->  (
( f `  B
)  =  ( F `
 ( f  |`  B ) )  -> 
( f  C_ recs ( F )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) )
3837exp31 587 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  Fn  x  ->  (
x  e.  On  ->  ( B  e.  dom  f  ->  ( ( f `  B )  =  ( F `  ( f  |`  B ) )  -> 
( f  C_ recs ( F )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) ) ) )
3938com34 77 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  Fn  x  ->  (
x  e.  On  ->  ( ( f `  B
)  =  ( F `
 ( f  |`  B ) )  -> 
( B  e.  dom  f  ->  ( f  C_ recs ( F )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) ) ) )
4039com24 81 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  Fn  x  ->  ( B  e.  dom  f  -> 
( ( f `  B )  =  ( F `  ( f  |`  B ) )  -> 
( x  e.  On  ->  ( f  C_ recs ( F )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) ) ) )
4122, 40sylbird 226 . . . . . . . . . . . . . . . . . . 19  |-  ( f  Fn  x  ->  ( B  e.  x  ->  ( ( f `  B
)  =  ( F `
 ( f  |`  B ) )  -> 
( x  e.  On  ->  ( f  C_ recs ( F )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) ) ) )
4241com3l 75 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  x  ->  (
( f `  B
)  =  ( F `
 ( f  |`  B ) )  -> 
( f  Fn  x  ->  ( x  e.  On  ->  ( f  C_ recs ( F )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) ) ) )
4320, 42syld 40 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  x  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  -> 
( f  Fn  x  ->  ( x  e.  On  ->  ( f  C_ recs ( F )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) ) ) )
4443com24 81 . . . . . . . . . . . . . . . 16  |-  ( B  e.  x  ->  (
x  e.  On  ->  ( f  Fn  x  -> 
( A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )  -> 
( f  C_ recs ( F )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) ) ) )
4544imp4d 575 . . . . . . . . . . . . . . 15  |-  ( B  e.  x  ->  (
( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  ->  ( f  C_ recs
( F )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) ) )
4615, 45mpdi 38 . . . . . . . . . . . . . 14  |-  ( B  e.  x  ->  (
( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) )
477, 46syl 15 . . . . . . . . . . . . 13  |-  ( ( f  Fn  x  /\  <. B ,  z >.  e.  f )  ->  (
( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) )
4847exp4d 592 . . . . . . . . . . . 12  |-  ( ( f  Fn  x  /\  <. B ,  z >.  e.  f )  ->  (
x  e.  On  ->  ( f  Fn  x  -> 
( A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) ) ) )
4948ex 423 . . . . . . . . . . 11  |-  ( f  Fn  x  ->  ( <. B ,  z >.  e.  f  ->  ( x  e.  On  ->  (
f  Fn  x  -> 
( A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) ) ) ) )
5049com4r 80 . . . . . . . . . 10  |-  ( f  Fn  x  ->  (
f  Fn  x  -> 
( <. B ,  z
>.  e.  f  ->  (
x  e.  On  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) ) ) ) )
5150pm2.43i 43 . . . . . . . . 9  |-  ( f  Fn  x  ->  ( <. B ,  z >.  e.  f  ->  ( x  e.  On  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) ) ) )
5251com3l 75 . . . . . . . 8  |-  ( <. B ,  z >.  e.  f  ->  ( x  e.  On  ->  ( f  Fn  x  ->  ( A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) ) ) ) )
5352imp4a 572 . . . . . . 7  |-  ( <. B ,  z >.  e.  f  ->  ( x  e.  On  ->  ( (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) ) )
5453rexlimdv 2666 . . . . . 6  |-  ( <. B ,  z >.  e.  f  ->  ( E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) ) )
5554imp 418 . . . . 5  |-  ( (
<. B ,  z >.  e.  f  /\  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
5655exlimiv 1666 . . . 4  |-  ( E. f ( <. B , 
z >.  e.  f  /\  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) )  ->  (recs ( F ) `  B )  =  ( F `  (recs ( F )  |`  B ) ) )
576, 56sylbi 187 . . 3  |-  ( <. B ,  z >.  e. recs
( F )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) )
5857exlimiv 1666 . 2  |-  ( E. z <. B ,  z
>.  e. recs ( F )  ->  (recs ( F ) `  B )  =  ( F `  (recs ( F )  |`  B ) ) )
592, 58syl 15 1  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   <.cop 3643   U.cuni 3827   Oncon0 4392   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  tfrlem11  6404  tfr2a  6411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-recs 6388
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