MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem12 Structured version   Unicode version

Theorem tfrlem12 7050
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 7045 . . . . 5  |-  Ord  dom recs ( F )
32a1i 11 . . . 4  |-  (recs ( F )  e.  _V  ->  Ord  dom recs ( F
) )
4 dmexg 6704 . . . 4  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  _V )
5 elon2 4878 . . . 4  |-  ( dom recs
( F )  e.  On  <->  ( Ord  dom recs ( F )  /\  dom recs ( F )  e.  _V ) )
63, 4, 5sylanbrc 662 . . 3  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  On )
7 suceloni 6621 . . . 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 7048 . . . 4  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
101, 8tfrlem11 7049 . . . . . 6  |-  ( dom recs
( F )  e.  On  ->  ( z  e.  suc  dom recs ( F
)  ->  ( C `  z )  =  ( F `  ( C  |`  z ) ) ) )
1110ralrimiv 2866 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  A. z  e.  suc  dom recs ( F
) ( C `  z )  =  ( F `  ( C  |`  z ) ) )
12 fveq2 5848 . . . . . . 7  |-  ( z  =  y  ->  ( C `  z )  =  ( C `  y ) )
13 reseq2 5257 . . . . . . . 8  |-  ( z  =  y  ->  ( C  |`  z )  =  ( C  |`  y
) )
1413fveq2d 5852 . . . . . . 7  |-  ( z  =  y  ->  ( F `  ( C  |`  z ) )  =  ( F `  ( C  |`  y ) ) )
1512, 14eqeq12d 2476 . . . . . 6  |-  ( z  =  y  ->  (
( C `  z
)  =  ( F `
 ( C  |`  z ) )  <->  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
1615cbvralv 3081 . . . . 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 196 . . . 4  |-  ( dom recs
( F )  e.  On  ->  A. y  e.  suc  dom recs ( F
) ( C `  y )  =  ( F `  ( C  |`  y ) ) )
18 fneq2 5652 . . . . . 6  |-  ( x  =  suc  dom recs ( F )  ->  ( C  Fn  x  <->  C  Fn  suc  dom recs ( F ) ) )
19 raleq 3051 . . . . . 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 708 . . . . 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 3207 . . . 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 1224 . . 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 4678 . . . . 5  |-  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. }  e.  _V
25 unexg 6574 . . . . 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 669 . . . 4  |-  (recs ( F )  e.  _V  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  e.  _V )
278, 26syl5eqel 2546 . . 3  |-  (recs ( F )  e.  _V  ->  C  e.  _V )
28 fneq1 5651 . . . . . 6  |-  ( f  =  C  ->  (
f  Fn  x  <->  C  Fn  x ) )
29 fveq1 5847 . . . . . . . 8  |-  ( f  =  C  ->  (
f `  y )  =  ( C `  y ) )
30 reseq1 5256 . . . . . . . . 9  |-  ( f  =  C  ->  (
f  |`  y )  =  ( C  |`  y
) )
3130fveq2d 5852 . . . . . . . 8  |-  ( f  =  C  ->  ( F `  ( f  |`  y ) )  =  ( F `  ( C  |`  y ) ) )
3229, 31eqeq12d 2476 . . . . . . 7  |-  ( f  =  C  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
3332ralbidv 2893 . . . . . 6  |-  ( f  =  C  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  A. y  e.  x  ( C `  y )  =  ( F `  ( C  |`  y ) ) ) )
3428, 33anbi12d 708 . . . . 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 2965 . . . 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 3245 . . 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 232 1  |-  (recs ( F )  e.  _V  ->  C  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805   _Vcvv 3106    u. cun 3459   {csn 4016   <.cop 4022   Ord word 4866   Oncon0 4867   suc csuc 4869   dom cdm 4988    |` cres 4990    Fn wfn 5565   ` cfv 5570  recscrecs 7033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-recs 7034
This theorem is referenced by:  tfrlem13  7051
  Copyright terms: Public domain W3C validator