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Theorem tfrlem9a 6402
Description: Lemma for transfinite recursion. Without using ax-rep 4131, 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 6399 . . . 4  |-  Fun recs ( F )
3 funfvop 5637 . . . 4  |-  ( ( Fun recs ( F )  /\  B  e.  dom recs ( F ) )  ->  <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
) )
42, 3mpan 651 . . 3  |-  ( B  e.  dom recs ( F
)  ->  <. B , 
(recs ( F ) `
 B ) >.  e. recs ( F ) )
51recsfval 6397 . . . . 5  |- recs ( F )  =  U. A
65eleq2i 2347 . . . 4  |-  ( <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
)  <->  <. B ,  (recs ( F ) `  B ) >.  e.  U. A )
7 eluni 3830 . . . 4  |-  ( <. B ,  (recs ( F ) `  B
) >.  e.  U. A  <->  E. g ( <. B , 
(recs ( F ) `
 B ) >.  e.  g  /\  g  e.  A ) )
86, 7bitri 240 . . 3  |-  ( <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
)  <->  E. g ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)
94, 8sylib 188 . 2  |-  ( B  e.  dom recs ( F
)  ->  E. g
( <. B ,  (recs ( F ) `  B ) >.  e.  g  /\  g  e.  A
) )
10 simprr 733 . . . . . 6  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  g  e.  A )
111tfrlem3 6393 . . . . . . 7  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
1211abeq2i 2390 . . . . . 6  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
1310, 12sylib 188 . . . . 5  |-  ( ( 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 10 . . . . . . . . . 10  |-  ( ( ( 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 737 . . . . . . . . . . . 12  |-  ( ( ( 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 3855 . . . . . . . . . . . 12  |-  ( g  e.  A  ->  g  C_ 
U. A )
1715, 16syl 15 . . . . . . . . . . 11  |-  ( ( ( 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 3225 . . . . . . . . . 10  |-  ( ( ( 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 5343 . . . . . . . . . . . . . 14  |-  ( g  Fn  z  ->  dom  g  =  z )
2019ad2antll 709 . . . . . . . . . . . . 13  |-  ( ( ( 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 732 . . . . . . . . . . . . 13  |-  ( ( ( 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 2357 . . . . . . . . . . . 12  |-  ( ( ( 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 4402 . . . . . . . . . . . 12  |-  ( dom  g  e.  On  ->  Ord 
dom  g )
2422, 23syl 15 . . . . . . . . . . 11  |-  ( ( ( 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 730 . . . . . . . . . . . 12  |-  ( ( ( 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 5539 . . . . . . . . . . . . 13  |-  (recs ( F ) `  B
)  e.  _V
2726a1i 10 . . . . . . . . . . . 12  |-  ( ( ( 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 736 . . . . . . . . . . . . 13  |-  ( ( ( 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 4024 . . . . . . . . . . . . 13  |-  ( B g (recs ( F ) `  B )  <->  <. B ,  (recs ( F ) `  B
) >.  e.  g )
3028, 29sylibr 203 . . . . . . . . . . . 12  |-  ( ( ( 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 4884 . . . . . . . . . . . 12  |-  ( ( B  e.  dom recs ( F )  /\  (recs ( F ) `  B
)  e.  _V  /\  B g (recs ( F ) `  B
) )  ->  B  e.  dom  g )
3225, 27, 30, 31syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( 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 4408 . . . . . . . . . . 11  |-  ( ( Ord  dom  g  /\  B  e.  dom  g )  ->  B  C_  dom  g )
3424, 32, 33syl2anc 642 . . . . . . . . . 10  |-  ( ( ( 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 5295 . . . . . . . . . 10  |-  ( ( Fun recs ( F )  /\  g  C_ recs ( F )  /\  B  C_  dom  g )  ->  (recs ( F )  |`  B )  =  ( g  |`  B ) )
3614, 18, 34, 35syl3anc 1182 . . . . . . . . 9  |-  ( ( ( 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 2791 . . . . . . . . . . 11  |-  g  e. 
_V
3837resex 4995 . . . . . . . . . 10  |-  ( g  |`  B )  e.  _V
3938a1i 10 . . . . . . . . 9  |-  ( ( ( 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 2357 . . . . . . . 8  |-  ( ( ( 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 598 . . . . . . 7  |-  ( ( ( 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 454 . . . . . 6  |-  ( ( ( 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 2667 . . . . 5  |-  ( ( 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 14 . . . 4  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  (recs ( F )  |`  B )  e.  _V )
4544ex 423 . . 3  |-  ( B  e.  dom recs ( F
)  ->  ( ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )  ->  (recs ( F )  |`  B )  e.  _V ) )
4645exlimdv 1664 . 2  |-  ( B  e.  dom recs ( F
)  ->  ( E. g ( <. B , 
(recs ( F ) `
 B ) >.  e.  g  /\  g  e.  A )  ->  (recs ( F )  |`  B )  e.  _V ) )
479, 46mpd 14 1  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   <.cop 3643   U.cuni 3827   class class class wbr 4023   Ord word 4391   Oncon0 4392   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  tfrlem15  6408  tfrlem16  6409  rdgseg  6435
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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