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Theorem tfrlem9 6648
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 5068 . . 3  |-  ( B  e.  dom recs ( F
)  ->  ( B  e.  dom recs ( F )  <->  E. z <. B ,  z
>.  e. recs ( F ) ) )
21ibi 234 . 2  |-  ( B  e.  dom recs ( F
)  ->  E. z <. B ,  z >.  e. recs ( F ) )
3 df-recs 6635 . . . . . 6  |- recs ( F )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
43eleq2i 2502 . . . . 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 4029 . . . . 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 242 . . . 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 5550 . . . . . . . . . . . . . 14  |-  ( ( f  Fn  x  /\  <. B ,  z >.  e.  f )  ->  B  e.  x )
8 rspe 2769 . . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  <->  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
11 elssuni 4045 . . . . . . . . . . . . . . . . . 18  |-  ( f  e.  A  ->  f  C_ 
U. A )
129recsfval 6644 . . . . . . . . . . . . . . . . . 18  |- recs ( F )  =  U. A
1311, 12syl6sseqr 3397 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  ->  f  C_ recs
( F ) )
1410, 13sylbir 206 . . . . . . . . . . . . . . . 16  |-  ( E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) )  -> 
f  C_ recs ( F
) )
158, 14syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  ->  f  C_ recs ( F ) )
16 fveq2 5730 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  B  ->  (
f `  y )  =  ( f `  B ) )
17 reseq2 5143 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  B  ->  (
f  |`  y )  =  ( f  |`  B ) )
1817fveq2d 5734 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  B  ->  ( F `  ( f  |`  y ) )  =  ( F `  (
f  |`  B ) ) )
1916, 18eqeq12d 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  B  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( f `  B )  =  ( F `  ( f  |`  B ) ) ) )
2019rspcv 3050 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  x  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  -> 
( f `  B
)  =  ( F `
 ( f  |`  B ) ) ) )
21 fndm 5546 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  Fn  x  ->  dom  f  =  x )
2221eleq2d 2505 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  Fn  x  ->  ( B  e.  dom  f  <->  B  e.  x ) )
239tfrlem7 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  Fun recs ( F )
24 funssfv 5748 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  e. 
dom  f )  -> 
(recs ( F ) `
 B )  =  ( f `  B
) )
2523, 24mp3an1 1267 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f  C_ recs ( F
)  /\  B  e.  dom  f )  ->  (recs ( F ) `  B
)  =  ( f `
 B ) )
2625adantrl 698 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
(recs ( F ) `
 B )  =  ( f `  B
) )
2721eleq1d 2504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( f  Fn  x  ->  ( dom  f  e.  On  <->  x  e.  On ) )
28 onelss 4625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( dom  f  e.  On  ->  ( B  e.  dom  f  ->  B  C_  dom  f ) )
2927, 28syl6bir 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( f  Fn  x  ->  (
x  e.  On  ->  ( B  e.  dom  f  ->  B  C_  dom  f ) ) )
3029imp31 423 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f )  ->  B  C_ 
dom  f )
31 fun2ssres 5496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  C_  dom  f )  ->  (recs ( F )  |`  B )  =  ( f  |`  B ) )
3231fveq2d 5734 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  C_  dom  f )  ->  ( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3323, 32mp3an1 1267 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f  C_ recs ( F
)  /\  B  C_  dom  f )  ->  ( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3430, 33sylan2 462 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3526, 34eqeq12d 2452 . . . . . . . . . . . . . . . . . . . . . . . . 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 607 . . . . . . . . . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . . . . . . . . . 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 589 . . . . . . . . . . . . . . . . . . . . . 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 80 . . . . . . . . . . . . . . . . . . . . 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 84 . . . . . . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . . . . 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 43 . . . . . . . . . . . . . . . . 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 84 . . . . . . . . . . . . . . . 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 577 . . . . . . . . . . . . . . 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 41 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 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 594 . . . . . . . . . . . 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 425 . . . . . . . . . . 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 83 . . . . . . . . . 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 46 . . . . . . . . 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 78 . . . . . . . 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 574 . . . . . . 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 2831 . . . . . 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 420 . . . . 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 1645 . . . 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 189 . . 3  |-  ( <. B ,  z >.  e. recs
( F )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) )
5857exlimiv 1645 . 2  |-  ( E. z <. B ,  z
>.  e. recs ( F )  ->  (recs ( F ) `  B )  =  ( F `  (recs ( F )  |`  B ) ) )
592, 58syl 16 1  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708    C_ wss 3322   <.cop 3819   U.cuni 4017   Oncon0 4583   dom cdm 4880    |` cres 4882   Fun wfun 5450    Fn wfn 5451   ` cfv 5456  recscrecs 6634
This theorem is referenced by:  tfrlem11  6651  tfr2a  6658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-recs 6635
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