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Theorem tfrlem9a 6584
Description: Lemma for transfinite recursion. Without using ax-rep 4262, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
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
tfrlem9a  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
Distinct variable groups:    x, f,
y, B    f, F, x, y
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem9a
Dummy variables  g 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . 5  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem7 6581 . . . 4  |-  Fun recs ( F )
3 funfvop 5782 . . . 4  |-  ( ( Fun recs ( F )  /\  B  e.  dom recs ( F ) )  ->  <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
) )
42, 3mpan 652 . . 3  |-  ( B  e.  dom recs ( F
)  ->  <. B , 
(recs ( F ) `
 B ) >.  e. recs ( F ) )
51recsfval 6579 . . . . 5  |- recs ( F )  =  U. A
65eleq2i 2452 . . . 4  |-  ( <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
)  <->  <. B ,  (recs ( F ) `  B ) >.  e.  U. A )
7 eluni 3961 . . . 4  |-  ( <. B ,  (recs ( F ) `  B
) >.  e.  U. A  <->  E. g ( <. B , 
(recs ( F ) `
 B ) >.  e.  g  /\  g  e.  A ) )
86, 7bitri 241 . . 3  |-  ( <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
)  <->  E. g ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)
94, 8sylib 189 . 2  |-  ( B  e.  dom recs ( F
)  ->  E. g
( <. B ,  (recs ( F ) `  B ) >.  e.  g  /\  g  e.  A
) )
10 simprr 734 . . . 4  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  g  e.  A )
111tfrlem3 6575 . . . . 5  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
1211abeq2i 2495 . . . 4  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
1310, 12sylib 189 . . 3  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
142a1i 11 . . . . . . . 8  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  Fun recs ( F ) )
15 simplrr 738 . . . . . . . . . 10  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  -> 
g  e.  A )
16 elssuni 3986 . . . . . . . . . 10  |-  ( g  e.  A  ->  g  C_ 
U. A )
1715, 16syl 16 . . . . . . . . 9  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  -> 
g  C_  U. A )
1817, 5syl6sseqr 3339 . . . . . . . 8  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  -> 
g  C_ recs ( F
) )
19 fndm 5485 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  dom  g  =  z )
2019ad2antll 710 . . . . . . . . . . 11  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  dom  g  =  z
)
21 simprl 733 . . . . . . . . . . 11  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  -> 
z  e.  On )
2220, 21eqeltrd 2462 . . . . . . . . . 10  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  dom  g  e.  On )
23 eloni 4533 . . . . . . . . . 10  |-  ( dom  g  e.  On  ->  Ord 
dom  g )
2422, 23syl 16 . . . . . . . . 9  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  Ord  dom  g )
25 simpll 731 . . . . . . . . . 10  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  B  e.  dom recs ( F ) )
26 fvex 5683 . . . . . . . . . . 11  |-  (recs ( F ) `  B
)  e.  _V
2726a1i 11 . . . . . . . . . 10  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  -> 
(recs ( F ) `
 B )  e. 
_V )
28 simplrl 737 . . . . . . . . . . 11  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  <. B ,  (recs ( F ) `  B
) >.  e.  g )
29 df-br 4155 . . . . . . . . . . 11  |-  ( B g (recs ( F ) `  B )  <->  <. B ,  (recs ( F ) `  B
) >.  e.  g )
3028, 29sylibr 204 . . . . . . . . . 10  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  B g (recs ( F ) `  B
) )
31 breldmg 5016 . . . . . . . . . 10  |-  ( ( B  e.  dom recs ( F )  /\  (recs ( F ) `  B
)  e.  _V  /\  B g (recs ( F ) `  B
) )  ->  B  e.  dom  g )
3225, 27, 30, 31syl3anc 1184 . . . . . . . . 9  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  B  e.  dom  g )
33 ordelss 4539 . . . . . . . . 9  |-  ( ( Ord  dom  g  /\  B  e.  dom  g )  ->  B  C_  dom  g )
3424, 32, 33syl2anc 643 . . . . . . . 8  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  ->  B  C_  dom  g )
35 fun2ssres 5435 . . . . . . . 8  |-  ( ( Fun recs ( F )  /\  g  C_ recs ( F )  /\  B  C_  dom  g )  ->  (recs ( F )  |`  B )  =  ( g  |`  B ) )
3614, 18, 34, 35syl3anc 1184 . . . . . . 7  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  -> 
(recs ( F )  |`  B )  =  ( g  |`  B )
)
37 vex 2903 . . . . . . . . 9  |-  g  e. 
_V
3837resex 5127 . . . . . . . 8  |-  ( g  |`  B )  e.  _V
3938a1i 11 . . . . . . 7  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  -> 
( g  |`  B )  e.  _V )
4036, 39eqeltrd 2462 . . . . . 6  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  ( z  e.  On  /\  g  Fn  z ) )  -> 
(recs ( F )  |`  B )  e.  _V )
4140expr 599 . . . . 5  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  z  e.  On )  ->  ( g  Fn  z  ->  (recs ( F )  |`  B )  e.  _V ) )
4241adantrd 455 . . . 4  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  z  e.  On )  ->  ( ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  -> 
(recs ( F )  |`  B )  e.  _V ) )
4342rexlimdva 2774 . . 3  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  ( E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  -> 
(recs ( F )  |`  B )  e.  _V ) )
4413, 43mpd 15 . 2  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  (recs ( F )  |`  B )  e.  _V )
459, 44exlimddv 1645 1  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   A.wral 2650   E.wrex 2651   _Vcvv 2900    C_ wss 3264   <.cop 3761   U.cuni 3958   class class class wbr 4154   Ord word 4522   Oncon0 4523   dom cdm 4819    |` cres 4821   Fun wfun 5389    Fn wfn 5390   ` cfv 5395  recscrecs 6569
This theorem is referenced by:  tfrlem15  6590  tfrlem16  6591  rdgseg  6617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403  df-recs 6570
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