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Theorem tfrlem9a 7052
Description: Lemma for transfinite recursion. Without using ax-rep 4558, 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 7049 . . . 4  |-  Fun recs ( F )
3 funfvop 5991 . . . 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 7047 . . . . 5  |- recs ( F )  =  U. A
65eleq2i 2545 . . . 4  |-  ( <. B ,  (recs ( F ) `  B
) >.  e. recs ( F
)  <->  <. B ,  (recs ( F ) `  B ) >.  e.  U. A )
7 eluni 4248 . . . 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 3116 . . . . 5  |-  g  e. 
_V
121, 11tfrlem3a 7043 . . . 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 4275 . . . . . . . . . 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 3551 . . . . . . . 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 5678 . . . . . . . . . . . 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 2555 . . . . . . . . . 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 4888 . . . . . . . . . 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 5874 . . . . . . . . . . 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 4448 . . . . . . . . . . 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 5206 . . . . . . . . . 10  |-  ( ( B  e.  dom recs ( F )  /\  (recs ( F ) `  B
)  e.  _V  /\  B g (recs ( F ) `  B
) )  ->  B  e.  dom  g )
3225, 27, 30, 31syl3anc 1228 . . . . . . . . 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 4894 . . . . . . . . 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 5627 . . . . . . . 8  |-  ( ( Fun recs ( F )  /\  g  C_ recs ( F )  /\  B  C_  dom  g )  ->  (recs ( F )  |`  B )  =  ( g  |`  B ) )
3614, 18, 34, 35syl3anc 1228 . . . . . . 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 5315 . . . . . . . 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 2555 . . . . . 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 2955 . . 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 1702 1  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   <.cop 4033   U.cuni 4245   class class class wbr 4447   Ord word 4877   Oncon0 4878   dom cdm 4999    |` cres 5001   Fun wfun 5580    Fn wfn 5581   ` cfv 5586  recscrecs 7038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-recs 7039
This theorem is referenced by:  tfrlem15  7058  tfrlem16  7059  rdgseg  7085
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