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Theorem tfrlem9a 6942
Description: Lemma for transfinite recursion. Without using ax-rep 4498, 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  a 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 6939 . . . 4  |-  Fun recs ( F )
3 funfvop 5911 . . . 4  |-  ( ( Fun recs ( F )  /\  B  e.  dom recs ( F ) )  ->  <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
) )
42, 3mpan 670 . . 3  |-  ( B  e.  dom recs ( F
)  ->  <. B , 
(recs ( F ) `
 B ) >.  e. recs ( F ) )
51recsfval 6937 . . . . 5  |- recs ( F )  =  U. A
65eleq2i 2527 . . . 4  |-  ( <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
)  <->  <. B ,  (recs ( F ) `  B ) >.  e.  U. A )
7 eluni 4189 . . . 4  |-  ( <. B ,  (recs ( F ) `  B
) >.  e.  U. A  <->  E. g ( <. B , 
(recs ( F ) `
 B ) >.  e.  g  /\  g  e.  A ) )
86, 7bitri 249 . . 3  |-  ( <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
)  <->  E. g ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)
94, 8sylib 196 . 2  |-  ( B  e.  dom recs ( F
)  ->  E. g
( <. B ,  (recs ( F ) `  B ) >.  e.  g  /\  g  e.  A
) )
10 simprr 756 . . . 4  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  g  e.  A )
11 vex 3068 . . . . 5  |-  g  e. 
_V
121, 11tfrlem3a 6933 . . . 4  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a ) ) ) )
1310, 12sylib 196 . . 3  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  E. z  e.  On  ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a ) ) ) )
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 760 . . . . . . . . . 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 4216 . . . . . . . . . 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 3498 . . . . . . . 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 5605 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  dom  g  =  z )
2019ad2antll 728 . . . . . . . . . . 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 755 . . . . . . . . . . 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 2537 . . . . . . . . . 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 4824 . . . . . . . . . 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 753 . . . . . . . . . 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 5796 . . . . . . . . . . 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 759 . . . . . . . . . . 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 4388 . . . . . . . . . . 11  |-  ( B g (recs ( F ) `  B )  <->  <. B ,  (recs ( F ) `  B
) >.  e.  g )
3028, 29sylibr 212 . . . . . . . . . 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 5140 . . . . . . . . . 10  |-  ( ( B  e.  dom recs ( F )  /\  (recs ( F ) `  B
)  e.  _V  /\  B g (recs ( F ) `  B
) )  ->  B  e.  dom  g )
3225, 27, 30, 31syl3anc 1219 . . . . . . . . 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 4830 . . . . . . . . 9  |-  ( ( Ord  dom  g  /\  B  e.  dom  g )  ->  B  C_  dom  g )
3424, 32, 33syl2anc 661 . . . . . . . 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 5554 . . . . . . . 8  |-  ( ( Fun recs ( F )  /\  g  C_ recs ( F )  /\  B  C_  dom  g )  ->  (recs ( F )  |`  B )  =  ( g  |`  B ) )
3614, 18, 34, 35syl3anc 1219 . . . . . . 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 )
)
3711resex 5245 . . . . . . . 8  |-  ( g  |`  B )  e.  _V
3837a1i 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 )
3936, 38eqeltrd 2537 . . . . . 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 )
4039expr 615 . . . . 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 ) )
4140adantrd 468 . . . 4  |-  ( ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  /\  z  e.  On )  ->  ( ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a
) ) )  -> 
(recs ( F )  |`  B )  e.  _V ) )
4241rexlimdva 2934 . . 3  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  ( E. z  e.  On  (
g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a
) ) )  -> 
(recs ( F )  |`  B )  e.  _V ) )
4313, 42mpd 15 . 2  |-  ( ( B  e.  dom recs ( F )  /\  ( <. B ,  (recs ( F ) `  B
) >.  e.  g  /\  g  e.  A )
)  ->  (recs ( F )  |`  B )  e.  _V )
449, 43exlimddv 1693 1  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   A.wral 2793   E.wrex 2794   _Vcvv 3065    C_ wss 3423   <.cop 3978   U.cuni 4186   class class class wbr 4387   Ord word 4813   Oncon0 4814   dom cdm 4935    |` cres 4937   Fun wfun 5507    Fn wfn 5508   ` cfv 5513  recscrecs 6928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-fv 5521  df-recs 6929
This theorem is referenced by:  tfrlem15  6948  tfrlem16  6949  rdgseg  6975
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