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Theorem tfrlem15 6856
Description: Lemma for transfinite recursion. Without assuming ax-rep 4408, 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 6850 . . 3  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
32adantl 466 . 2  |-  ( ( B  e.  On  /\  B  e.  dom recs ( F ) )  ->  (recs ( F )  |`  B )  e.  _V )
41tfrlem13 6854 . . . 4  |-  -. recs ( F )  e.  _V
5 simpr 461 . . . . 5  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  (recs ( F )  |`  B )  e.  _V )
6 resss 5139 . . . . . . . 8  |-  (recs ( F )  |`  B ) 
C_ recs ( F )
76a1i 11 . . . . . . 7  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  B )  C_ recs
( F ) )
81tfrlem6 6846 . . . . . . . . 9  |-  Rel recs ( F )
9 resdm 5153 . . . . . . . . 9  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
108, 9ax-mp 5 . . . . . . . 8  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
11 ssres2 5142 . . . . . . . 8  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  dom recs ( F ) )  C_  (recs ( F )  |`  B ) )
1210, 11syl5eqssr 3406 . . . . . . 7  |-  ( dom recs
( F )  C_  B  -> recs ( F ) 
C_  (recs ( F )  |`  B )
)
137, 12eqssd 3378 . . . . . 6  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  B )  = recs ( F ) )
1413eleq1d 2509 . . . . 5  |-  ( dom recs
( F )  C_  B  ->  ( (recs ( F )  |`  B )  e.  _V  <-> recs ( F
)  e.  _V )
)
155, 14syl5ibcom 220 . . . 4  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  ( dom recs ( F )  C_  B  -> recs ( F )  e. 
_V ) )
164, 15mtoi 178 . . 3  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  -.  dom recs ( F )  C_  B )
171tfrlem8 6848 . . . 4  |-  Ord  dom recs ( F )
18 eloni 4734 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
1918adantr 465 . . . 4  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  Ord  B )
20 ordtri1 4757 . . . . 5  |-  ( ( Ord  dom recs ( F
)  /\  Ord  B )  ->  ( dom recs ( F )  C_  B  <->  -.  B  e.  dom recs ( F ) ) )
2120con2bid 329 . . . 4  |-  ( ( Ord  dom recs ( F
)  /\  Ord  B )  ->  ( B  e. 
dom recs ( F )  <->  -.  dom recs ( F )  C_  B
) )
2217, 19, 21sylancr 663 . . 3  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  ( B  e. 
dom recs ( F )  <->  -.  dom recs ( F )  C_  B
) )
2316, 22mpbird 232 . 2  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  B  e.  dom recs ( F ) )
243, 23impbida 828 1  |-  ( B  e.  On  ->  ( B  e.  dom recs ( F )  <->  (recs ( F )  |`  B )  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   Ord word 4723   Oncon0 4724   dom cdm 4845    |` cres 4847   Rel wrel 4850    Fn wfn 5418   ` cfv 5423  recscrecs 6836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431  df-recs 6837
This theorem is referenced by:  tfrlem16  6857  tfr2b  6860
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