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Theorem tfrlem9 7009
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 5139 . . 3  |-  ( B  e.  dom recs ( F
)  ->  ( B  e.  dom recs ( F )  <->  E. z <. B ,  z
>.  e. recs ( F ) ) )
21ibi 241 . 2  |-  ( B  e.  dom recs ( F
)  ->  E. z <. B ,  z >.  e. recs ( F ) )
3 df-recs 6997 . . . . . 6  |- recs ( F )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
43eleq2i 2478 . . . . 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 4199 . . . . 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 249 . . . 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 5619 . . . . . . . . . . . . . 14  |-  ( ( f  Fn  x  /\  <. B ,  z >.  e.  f )  ->  B  e.  x )
8 rspe 2859 . . . . . . . . . . . . . . . 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 2527 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  <->  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
11 elssuni 4217 . . . . . . . . . . . . . . . . . 18  |-  ( f  e.  A  ->  f  C_ 
U. A )
129recsfval 7005 . . . . . . . . . . . . . . . . . 18  |- recs ( F )  =  U. A
1311, 12syl6sseqr 3486 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  ->  f  C_ recs
( F ) )
1410, 13sylbir 213 . . . . . . . . . . . . . . . 16  |-  ( E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) )  -> 
f  C_ recs ( F
) )
158, 14syl 17 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  ->  f  C_ recs ( F ) )
16 fveq2 5803 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  B  ->  (
f `  y )  =  ( f `  B ) )
17 reseq2 5208 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  B  ->  (
f  |`  y )  =  ( f  |`  B ) )
1817fveq2d 5807 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  B  ->  ( F `  ( f  |`  y ) )  =  ( F `  (
f  |`  B ) ) )
1916, 18eqeq12d 2422 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  B  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( f `  B )  =  ( F `  ( f  |`  B ) ) ) )
2019rspcv 3153 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  x  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  -> 
( f `  B
)  =  ( F `
 ( f  |`  B ) ) ) )
21 fndm 5615 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  Fn  x  ->  dom  f  =  x )
2221eleq2d 2470 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  Fn  x  ->  ( B  e.  dom  f  <->  B  e.  x ) )
239tfrlem7 7007 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  Fun recs ( F )
24 funssfv 5818 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  e. 
dom  f )  -> 
(recs ( F ) `
 B )  =  ( f `  B
) )
2523, 24mp3an1 1311 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f  C_ recs ( F
)  /\  B  e.  dom  f )  ->  (recs ( F ) `  B
)  =  ( f `
 B ) )
2625adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
(recs ( F ) `
 B )  =  ( f `  B
) )
2721eleq1d 2469 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( f  Fn  x  ->  ( dom  f  e.  On  <->  x  e.  On ) )
28 onelss 4861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( dom  f  e.  On  ->  ( B  e.  dom  f  ->  B  C_  dom  f ) )
2927, 28syl6bir 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( f  Fn  x  ->  (
x  e.  On  ->  ( B  e.  dom  f  ->  B  C_  dom  f ) ) )
3029imp31 430 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f )  ->  B  C_ 
dom  f )
31 fun2ssres 5564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  C_  dom  f )  ->  (recs ( F )  |`  B )  =  ( f  |`  B ) )
3231fveq2d 5807 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Fun recs ( F )  /\  f  C_ recs ( F )  /\  B  C_  dom  f )  ->  ( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3323, 32mp3an1 1311 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f  C_ recs ( F
)  /\  B  C_  dom  f )  ->  ( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3430, 33sylan2 472 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  C_ recs ( F
)  /\  ( (
f  Fn  x  /\  x  e.  On )  /\  B  e.  dom  f ) )  -> 
( F `  (recs ( F )  |`  B ) )  =  ( F `
 ( f  |`  B ) ) )
3526, 34eqeq12d 2422 . . . . . . . . . . . . . . . . . . . . . . . . 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 620 . . . . . . . . . . . . . . . . . . . . . . . 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 602 . . . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . . . 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 40 . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . 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 607 . . . . . . . . . . . 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 432 . . . . . . . . . . 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 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 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 587 . . . . . . 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 2891 . . . . . 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 427 . . . . 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 1741 . . . 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 195 . . 3  |-  ( <. B ,  z >.  e. recs
( F )  -> 
(recs ( F ) `
 B )  =  ( F `  (recs ( F )  |`  B ) ) )
5857exlimiv 1741 . 2  |-  ( E. z <. B ,  z
>.  e. recs ( F )  ->  (recs ( F ) `  B )  =  ( F `  (recs ( F )  |`  B ) ) )
592, 58syl 17 1  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F ) `  B
)  =  ( F `
 (recs ( F )  |`  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403   E.wex 1631    e. wcel 1840   {cab 2385   A.wral 2751   E.wrex 2752    C_ wss 3411   <.cop 3975   U.cuni 4188   Oncon0 4819   dom cdm 4940    |` cres 4942   Fun wfun 5517    Fn wfn 5518   ` cfv 5523  recscrecs 6996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-fv 5531  df-recs 6997
This theorem is referenced by:  tfrlem11  7012  tfr2a  7019
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