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Theorem tfrlem12 6609
Description: Lemma for transfinite recursion. Show  C is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlem.3  |-  C  =  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
Assertion
Ref Expression
tfrlem12  |-  (recs ( F )  e.  _V  ->  C  e.  A )
Distinct variable groups:    x, f,
y, C    f, F, x, y
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem12
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . . 6  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem8 6604 . . . . 5  |-  Ord  dom recs ( F )
32a1i 11 . . . 4  |-  (recs ( F )  e.  _V  ->  Ord  dom recs ( F
) )
4 dmexg 5089 . . . 4  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  _V )
5 elon2 4552 . . . 4  |-  ( dom recs
( F )  e.  On  <->  ( Ord  dom recs ( F )  /\  dom recs ( F )  e.  _V ) )
63, 4, 5sylanbrc 646 . . 3  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  On )
7 suceloni 4752 . . . 4  |-  ( dom recs
( F )  e.  On  ->  suc  dom recs ( F )  e.  On )
8 tfrlem.3 . . . . 5  |-  C  =  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
91, 8tfrlem10 6607 . . . 4  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
101, 8tfrlem11 6608 . . . . . 6  |-  ( dom recs
( F )  e.  On  ->  ( z  e.  suc  dom recs ( F
)  ->  ( C `  z )  =  ( F `  ( C  |`  z ) ) ) )
1110ralrimiv 2748 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  A. z  e.  suc  dom recs ( F
) ( C `  z )  =  ( F `  ( C  |`  z ) ) )
12 fveq2 5687 . . . . . . 7  |-  ( z  =  y  ->  ( C `  z )  =  ( C `  y ) )
13 reseq2 5100 . . . . . . . 8  |-  ( z  =  y  ->  ( C  |`  z )  =  ( C  |`  y
) )
1413fveq2d 5691 . . . . . . 7  |-  ( z  =  y  ->  ( F `  ( C  |`  z ) )  =  ( F `  ( C  |`  y ) ) )
1512, 14eqeq12d 2418 . . . . . 6  |-  ( z  =  y  ->  (
( C `  z
)  =  ( F `
 ( C  |`  z ) )  <->  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
1615cbvralv 2892 . . . . 5  |-  ( A. z  e.  suc  dom recs ( F ) ( C `
 z )  =  ( F `  ( C  |`  z ) )  <->  A. y  e.  suc  dom recs
( F ) ( C `  y )  =  ( F `  ( C  |`  y ) ) )
1711, 16sylib 189 . . . 4  |-  ( dom recs
( F )  e.  On  ->  A. y  e.  suc  dom recs ( F
) ( C `  y )  =  ( F `  ( C  |`  y ) ) )
18 fneq2 5494 . . . . . 6  |-  ( x  =  suc  dom recs ( F )  ->  ( C  Fn  x  <->  C  Fn  suc  dom recs ( F ) ) )
19 raleq 2864 . . . . . 6  |-  ( x  =  suc  dom recs ( F )  ->  ( A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) )  <->  A. y  e.  suc  dom recs
( F ) ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
2018, 19anbi12d 692 . . . . 5  |-  ( x  =  suc  dom recs ( F )  ->  (
( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) )  <->  ( C  Fn  suc  dom recs ( F
)  /\  A. y  e.  suc  dom recs ( F
) ( C `  y )  =  ( F `  ( C  |`  y ) ) ) ) )
2120rspcev 3012 . . . 4  |-  ( ( suc  dom recs ( F
)  e.  On  /\  ( C  Fn  suc  dom recs
( F )  /\  A. y  e.  suc  dom recs ( F ) ( C `
 y )  =  ( F `  ( C  |`  y ) ) ) )  ->  E. x  e.  On  ( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
227, 9, 17, 21syl12anc 1182 . . 3  |-  ( dom recs
( F )  e.  On  ->  E. x  e.  On  ( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
236, 22syl 16 . 2  |-  (recs ( F )  e.  _V  ->  E. x  e.  On  ( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
24 snex 4365 . . . . 5  |-  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. }  e.  _V
25 unexg 4669 . . . . 5  |-  ( (recs ( F )  e. 
_V  /\  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. }  e.  _V )  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  e.  _V )
2624, 25mpan2 653 . . . 4  |-  (recs ( F )  e.  _V  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  e.  _V )
278, 26syl5eqel 2488 . . 3  |-  (recs ( F )  e.  _V  ->  C  e.  _V )
28 fneq1 5493 . . . . . 6  |-  ( f  =  C  ->  (
f  Fn  x  <->  C  Fn  x ) )
29 fveq1 5686 . . . . . . . 8  |-  ( f  =  C  ->  (
f `  y )  =  ( C `  y ) )
30 reseq1 5099 . . . . . . . . 9  |-  ( f  =  C  ->  (
f  |`  y )  =  ( C  |`  y
) )
3130fveq2d 5691 . . . . . . . 8  |-  ( f  =  C  ->  ( F `  ( f  |`  y ) )  =  ( F `  ( C  |`  y ) ) )
3229, 31eqeq12d 2418 . . . . . . 7  |-  ( f  =  C  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
3332ralbidv 2686 . . . . . 6  |-  ( f  =  C  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
3428, 33anbi12d 692 . . . . 5  |-  ( f  =  C  ->  (
( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) )  <-> 
( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) ) )
3534rexbidv 2687 . . . 4  |-  ( f  =  C  ->  ( E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) )  <->  E. x  e.  On  ( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) ) )
3635, 1elab2g 3044 . . 3  |-  ( C  e.  _V  ->  ( C  e.  A  <->  E. x  e.  On  ( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) ) )
3727, 36syl 16 . 2  |-  (recs ( F )  e.  _V  ->  ( C  e.  A  <->  E. x  e.  On  ( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) ) )
3823, 37mpbird 224 1  |-  (recs ( F )  e.  _V  ->  C  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916    u. cun 3278   {csn 3774   <.cop 3777   Ord word 4540   Oncon0 4541   suc csuc 4543   dom cdm 4837    |` cres 4839    Fn wfn 5408   ` cfv 5413  recscrecs 6591
This theorem is referenced by:  tfrlem13  6610
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-recs 6592
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