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Theorem tfrlem15 6612
Description: Lemma for transfinite recursion. Without assuming ax-rep 4280, 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 6606 . . 3  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
32adantl 453 . 2  |-  ( ( B  e.  On  /\  B  e.  dom recs ( F ) )  ->  (recs ( F )  |`  B )  e.  _V )
41tfrlem13 6610 . . . 4  |-  -. recs ( F )  e.  _V
5 simpr 448 . . . . 5  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  (recs ( F )  |`  B )  e.  _V )
6 resss 5129 . . . . . . . 8  |-  (recs ( F )  |`  B ) 
C_ recs ( F )
76a1i 11 . . . . . . 7  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  B )  C_ recs
( F ) )
81tfrlem6 6602 . . . . . . . . 9  |-  Rel recs ( F )
9 resdm 5143 . . . . . . . . 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 5132 . . . . . . . 8  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  dom recs ( F ) )  C_  (recs ( F )  |`  B ) )
1210, 11syl5eqssr 3353 . . . . . . 7  |-  ( dom recs
( F )  C_  B  -> recs ( F ) 
C_  (recs ( F )  |`  B )
)
137, 12eqssd 3325 . . . . . 6  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  B )  = recs ( F ) )
1413eleq1d 2470 . . . . 5  |-  ( dom recs
( F )  C_  B  ->  ( (recs ( F )  |`  B )  e.  _V  <-> recs ( F
)  e.  _V )
)
155, 14syl5ibcom 212 . . . 4  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  ( dom recs ( F )  C_  B  -> recs ( F )  e. 
_V ) )
164, 15mtoi 171 . . 3  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  -.  dom recs ( F )  C_  B )
171tfrlem8 6604 . . . 4  |-  Ord  dom recs ( F )
18 eloni 4551 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
1918adantr 452 . . . 4  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  Ord  B )
20 ordtri1 4574 . . . . 5  |-  ( ( Ord  dom recs ( F
)  /\  Ord  B )  ->  ( dom recs ( F )  C_  B  <->  -.  B  e.  dom recs ( F ) ) )
2120con2bid 320 . . . 4  |-  ( ( Ord  dom recs ( F
)  /\  Ord  B )  ->  ( B  e. 
dom recs ( F )  <->  -.  dom recs ( F )  C_  B
) )
2217, 19, 21sylancr 645 . . 3  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  ( B  e. 
dom recs ( F )  <->  -.  dom recs ( F )  C_  B
) )
2316, 22mpbird 224 . 2  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  B  e.  dom recs ( F ) )
243, 23impbida 806 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   Ord word 4540   Oncon0 4541   dom cdm 4837    |` cres 4839   Rel wrel 4842    Fn wfn 5408   ` cfv 5413  recscrecs 6591
This theorem is referenced by:  tfrlem16  6613  tfr2b  6616
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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