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Theorem tfrlem15 6294
Description: Lemma for transfinite recursion. Without assuming ax-rep 4028, 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 6288 . . 3  |-  ( B  e.  dom recs ( F
)  ->  (recs ( F )  |`  B )  e.  _V )
32adantl 454 . 2  |-  ( ( B  e.  On  /\  B  e.  dom recs ( F ) )  ->  (recs ( F )  |`  B )  e.  _V )
41tfrlem13 6292 . . . 4  |-  -. recs ( F )  e.  _V
5 simpr 449 . . . . 5  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  (recs ( F )  |`  B )  e.  _V )
6 resss 4886 . . . . . . . 8  |-  (recs ( F )  |`  B ) 
C_ recs ( F )
76a1i 12 . . . . . . 7  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  B )  C_ recs
( F ) )
81tfrlem6 6284 . . . . . . . . 9  |-  Rel recs ( F )
9 resdm 4900 . . . . . . . . 9  |-  ( Rel recs
( F )  -> 
(recs ( F )  |`  dom recs ( F ) )  = recs ( F ) )
108, 9ax-mp 10 . . . . . . . 8  |-  (recs ( F )  |`  dom recs ( F ) )  = recs ( F )
11 ssres2 4889 . . . . . . . 8  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  dom recs ( F ) )  C_  (recs ( F )  |`  B ) )
1210, 11syl5eqssr 3144 . . . . . . 7  |-  ( dom recs
( F )  C_  B  -> recs ( F ) 
C_  (recs ( F )  |`  B )
)
137, 12eqssd 3117 . . . . . 6  |-  ( dom recs
( F )  C_  B  ->  (recs ( F )  |`  B )  = recs ( F ) )
1413eleq1d 2319 . . . . 5  |-  ( dom recs
( F )  C_  B  ->  ( (recs ( F )  |`  B )  e.  _V  <-> recs ( F
)  e.  _V )
)
155, 14syl5ibcom 213 . . . 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 6286 . . . 4  |-  Ord  dom recs ( F )
18 eloni 4295 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
1918adantr 453 . . . 4  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  Ord  B )
20 ordtri1 4318 . . . . 5  |-  ( ( Ord  dom recs ( F
)  /\  Ord  B )  ->  ( dom recs ( F )  C_  B  <->  -.  B  e.  dom recs ( F ) ) )
2120con2bid 321 . . . 4  |-  ( ( Ord  dom recs ( F
)  /\  Ord  B )  ->  ( B  e. 
dom recs ( F )  <->  -.  dom recs ( F )  C_  B
) )
2217, 19, 21sylancr 647 . . 3  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  ( B  e. 
dom recs ( F )  <->  -.  dom recs ( F )  C_  B
) )
2316, 22mpbird 225 . 2  |-  ( ( B  e.  On  /\  (recs ( F )  |`  B )  e.  _V )  ->  B  e.  dom recs ( F ) )
243, 23impbida 808 1  |-  ( B  e.  On  ->  ( B  e.  dom recs ( F )  <->  (recs ( F )  |`  B )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2239   A.wral 2509   E.wrex 2510   _Vcvv 2727    C_ wss 3078   Ord word 4284   Oncon0 4285   dom cdm 4580    |` cres 4582   Rel wrel 4585    Fn wfn 4587   ` cfv 4592  recscrecs 6273
This theorem is referenced by:  tfrlem16  6295  tfr2b  6298
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608  df-recs 6274
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