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Theorem tfrlem12 6405
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 6400 . . . . 5  |-  Ord  dom recs ( F )
32a1i 10 . . . 4  |-  (recs ( F )  e.  _V  ->  Ord  dom recs ( F
) )
4 dmexg 4939 . . . 4  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  _V )
5 elon2 4403 . . . 4  |-  ( dom recs
( F )  e.  On  <->  ( Ord  dom recs ( F )  /\  dom recs ( F )  e.  _V ) )
63, 4, 5sylanbrc 645 . . 3  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  On )
7 suceloni 4604 . . . 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 6403 . . . 4  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
101, 8tfrlem11 6404 . . . . . 6  |-  ( dom recs
( F )  e.  On  ->  ( z  e.  suc  dom recs ( F
)  ->  ( C `  z )  =  ( F `  ( C  |`  z ) ) ) )
1110ralrimiv 2625 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  A. z  e.  suc  dom recs ( F
) ( C `  z )  =  ( F `  ( C  |`  z ) ) )
12 fveq2 5525 . . . . . . 7  |-  ( z  =  y  ->  ( C `  z )  =  ( C `  y ) )
13 reseq2 4950 . . . . . . . 8  |-  ( z  =  y  ->  ( C  |`  z )  =  ( C  |`  y
) )
1413fveq2d 5529 . . . . . . 7  |-  ( z  =  y  ->  ( F `  ( C  |`  z ) )  =  ( F `  ( C  |`  y ) ) )
1512, 14eqeq12d 2297 . . . . . 6  |-  ( z  =  y  ->  (
( C `  z
)  =  ( F `
 ( C  |`  z ) )  <->  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
1615cbvralv 2764 . . . . 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 188 . . . 4  |-  ( dom recs
( F )  e.  On  ->  A. y  e.  suc  dom recs ( F
) ( C `  y )  =  ( F `  ( C  |`  y ) ) )
18 fneq2 5334 . . . . . 6  |-  ( x  =  suc  dom recs ( F )  ->  ( C  Fn  x  <->  C  Fn  suc  dom recs ( F ) ) )
19 raleq 2736 . . . . . 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 691 . . . . 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 2884 . . . 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 1180 . . 3  |-  ( dom recs
( F )  e.  On  ->  E. x  e.  On  ( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
236, 22syl 15 . 2  |-  (recs ( F )  e.  _V  ->  E. x  e.  On  ( C  Fn  x  /\  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
24 snex 4216 . . . . 5  |-  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. }  e.  _V
25 unexg 4521 . . . . 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 652 . . . 4  |-  (recs ( F )  e.  _V  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  e.  _V )
278, 26syl5eqel 2367 . . 3  |-  (recs ( F )  e.  _V  ->  C  e.  _V )
28 fneq1 5333 . . . . . 6  |-  ( f  =  C  ->  (
f  Fn  x  <->  C  Fn  x ) )
29 fveq1 5524 . . . . . . . 8  |-  ( f  =  C  ->  (
f `  y )  =  ( C `  y ) )
30 reseq1 4949 . . . . . . . . 9  |-  ( f  =  C  ->  (
f  |`  y )  =  ( C  |`  y
) )
3130fveq2d 5529 . . . . . . . 8  |-  ( f  =  C  ->  ( F `  ( f  |`  y ) )  =  ( F `  ( C  |`  y ) ) )
3229, 31eqeq12d 2297 . . . . . . 7  |-  ( f  =  C  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
3332ralbidv 2563 . . . . . 6  |-  ( f  =  C  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
3428, 33anbi12d 691 . . . . 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 2564 . . . 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 2916 . . 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 15 . 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 223 1  |-  (recs ( F )  e.  _V  ->  C  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150   {csn 3640   <.cop 3643   Ord word 4391   Oncon0 4392   suc csuc 4394   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  tfrlem13  6406
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-suc 4398  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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