MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem15 Unicode version

Theorem tfrlem15 6408
Description: Lemma for transfinite recursion. Without assuming ax-rep 4131, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 14-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
tfrlem15  |-  ( B  e.  On  ->  ( 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 tfrlem15
StepHypRef Expression
1 tfrlem.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem9a 6402 . . 3  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
32adantl 452 . 2  |-  ( ( B  e.  On  /\  B  e.  dom recs ( F ) )  ->  (recs ( F )  |`  B )  e.  _V )
41tfrlem13 6406 . . . 4  |-  -. recs ( F )  e.  _V
5 simpr 447 . . . . 5  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  (recs ( F )  |`  B )  e.  _V )
6 resss 4979 . . . . . . . 8  |-  (recs ( F )  |`  B ) 
C_ recs ( F )
76a1i 10 . . . . . . 7  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  B )  C_ recs
( F ) )
81tfrlem6 6398 . . . . . . . . 9  |-  Rel recs ( F )
9 resdm 4993 . . . . . . . . 9  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
108, 9ax-mp 8 . . . . . . . 8  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
11 ssres2 4982 . . . . . . . 8  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  dom recs ( F ) )  C_  (recs ( F )  |`  B ) )
1210, 11syl5eqssr 3223 . . . . . . 7  |-  ( dom recs
( F )  C_  B  -> recs ( F ) 
C_  (recs ( F )  |`  B )
)
137, 12eqssd 3196 . . . . . 6  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  B )  = recs ( F ) )
1413eleq1d 2349 . . . . 5  |-  ( dom recs
( F )  C_  B  ->  ( (recs ( F )  |`  B )  e.  _V  <-> recs ( F
)  e.  _V )
)
155, 14syl5ibcom 211 . . . 4  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  ( dom recs ( F )  C_  B  -> recs ( F )  e. 
_V ) )
164, 15mtoi 169 . . 3  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  -.  dom recs ( F )  C_  B )
171tfrlem8 6400 . . . 4  |-  Ord  dom recs ( F )
18 eloni 4402 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
1918adantr 451 . . . 4  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  Ord  B )
20 ordtri1 4425 . . . . 5  |-  ( ( Ord  dom recs ( F
)  /\  Ord  B )  ->  ( dom recs ( F )  C_  B  <->  -.  B  e.  dom recs ( F ) ) )
2120con2bid 319 . . . 4  |-  ( ( Ord  dom recs ( F
)  /\  Ord  B )  ->  ( B  e. 
dom recs ( F )  <->  -.  dom recs ( F )  C_  B
) )
2217, 19, 21sylancr 644 . . 3  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  ( B  e. 
dom recs ( F )  <->  -.  dom recs ( F )  C_  B
) )
2316, 22mpbird 223 . 2  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  B  e.  dom recs ( F ) )
243, 23impbida 805 1  |-  ( B  e.  On  ->  ( B  e.  dom recs ( F )  <->  (recs ( F )  |`  B )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   Ord word 4391   Oncon0 4392   dom cdm 4689    |` cres 4691   Rel wrel 4694    Fn wfn 5250   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  tfrlem16  6409  tfr2b  6412
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
  Copyright terms: Public domain W3C validator