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Theorem tfrlem9 6310
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 4816 . . 3  |-  ( B  e.  dom recs ( F
)  ->  ( B  e.  dom recs ( F )  <->  E. z <. B ,  z
>.  e. recs ( F ) ) )
21ibi 176 . 2  |-  ( B  e.  dom recs ( F
)  ->  E. z <. B ,  z >.  e. recs ( F ) )
3 df-recs 6296 . . . . . 6  |- recs ( F )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
43eleq2i 2242 . . . . 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 3817 . . . . 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 184 . . . 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 5311 . . . . . . . . . . . . . 14  |-  ( ( f  Fn  x  /\  <. B ,  z >.  e.  f )  ->  B  e.  x )
8 rspe 2524 . . . . . . . . . . . . . . . 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 2286 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  <->  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
11 elssuni 3833 . . . . . . . . . . . . . . . . . 18  |-  ( f  e.  A  ->  f  C_ 
U. A )
129recsfval 6306 . . . . . . . . . . . . . . . . . 18  |- recs ( F )  =  U. A
1311, 12sseqtrrdi 3202 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  ->  f  C_ recs
( F ) )
1410, 13sylbir 135 . . . . . . . . . . . . . . . 16  |-  ( E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) )  -> 
f  C_ recs ( F
) )
158, 14syl 14 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  ->  f  C_ recs ( F ) )
16 fveq2 5507 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  B  ->  (
f `  y )  =  ( f `  B ) )
17 reseq2 4895 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  B  ->  (
f  |`  y )  =  ( f  |`  B ) )
1817fveq2d 5511 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  B  ->  ( F `  ( f  |`  y ) )  =  ( F `  (
f  |`  B ) ) )
1916, 18eqeq12d 2190 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  B  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( f `  B )  =  ( F `  ( f  |`  B ) ) ) )
2019rspcv 2835 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  x  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  -> 
( f `  B
)  =  ( F `
 ( f  |`  B ) ) ) )
21 fndm 5307 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  Fn  x  ->  dom  f  =  x )
2221eleq2d 2245 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  Fn  x  ->  ( B  e.  dom  f  <->  B  e.  x ) )
239tfrlem7 6308 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  Fun recs ( F )
24 funssfv 5533 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  e. 
dom  f )  -> 
(recs ( F ) `
 B )  =  ( f `  B
) )
2523, 24mp3an1 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f  C_ recs ( F
)  /\  B  e.  dom  f )  ->  (recs ( F ) `  B
)  =  ( f `
 B ) )
2625adantrl 478 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
(recs ( F ) `
 B )  =  ( f `  B
) )
2721eleq1d 2244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( f  Fn  x  ->  ( dom  f  e.  On  <->  x  e.  On ) )
28 onelss 4381 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( dom  f  e.  On  ->  ( B  e.  dom  f  ->  B  C_  dom  f ) )
2927, 28syl6bir 164 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( f  Fn  x  ->  (
x  e.  On  ->  ( B  e.  dom  f  ->  B  C_  dom  f ) ) )
3029imp31 256 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f )  ->  B  C_ 
dom  f )
31 fun2ssres 5251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  C_  dom  f )  ->  (recs ( F )  |`  B )  =  ( f  |`  B ) )
3231fveq2d 5511 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  C_  dom  f )  ->  ( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3323, 32mp3an1 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f  C_ recs ( F
)  /\  B  C_  dom  f )  ->  ( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3430, 33sylan2 286 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3526, 34eqeq12d 2190 . . . . . . . . . . . . . . . . . . . . . . . . 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 382 . . . . . . . . . . . . . . . . . . . . . . . 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 81 . . . . . . . . . . . . . . . . . . . . . . 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 364 . . . . . . . . . . . . . . . . . . . . . 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 83 . . . . . . . . . . . . . . . . . . . . 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 87 . . . . . . . . . . . . . . . . . . . 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 170 . . . . . . . . . . . . . . . . . . 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 81 . . . . . . . . . . . . . . . . . 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 45 . . . . . . . . . . . . . . . . 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 87 . . . . . . . . . . . . . . . 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 352 . . . . . . . . . . . . . . 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 43 . . . . . . . . . . . . . 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 14 . . . . . . . . . . . . 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 369 . . . . . . . . . . . 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 115 . . . . . . . . . . 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 86 . . . . . . . . . 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 49 . . . . . . . . 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 81 . . . . . . . 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 349 . . . . . . 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 2591 . . . . . 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 124 . . . . 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 1596 . . . 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 121 . . 3  |-  ( <. B ,  z >.  e. recs
( F )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) )
5857exlimiv 1596 . 2  |-  ( E. z <. B ,  z
>.  e. recs ( F )  ->  (recs ( F ) `  B )  =  ( F `  (recs ( F )  |`  B ) ) )
592, 58syl 14 1  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353   E.wex 1490    e. wcel 2146   {cab 2161   A.wral 2453   E.wrex 2454    C_ wss 3127   <.cop 3592   U.cuni 3805   Oncon0 4357   dom cdm 4620    |` cres 4622   Fun wfun 5202    Fn wfn 5203   ` cfv 5208  recscrecs 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-recs 6296
This theorem is referenced by:  tfr2a  6312  tfrlemiubacc  6321  tfr1onlemubacc  6337  tfrcllemubacc  6350
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