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Theorem tfr2b 6454
Description: Without assuming ax-rep 4168, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
tfr.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr2b  |-  ( Ord 
A  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )

Proof of Theorem tfr2b
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordeleqon 4617 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 eqid 2316 . . . . 5  |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
32tfrlem15 6450 . . . 4  |-  ( A  e.  On  ->  ( A  e.  dom recs ( G )  <->  (recs ( G )  |`  A )  e.  _V ) )
4 tfr.1 . . . . . 6  |-  F  = recs ( G )
54dmeqi 4917 . . . . 5  |-  dom  F  =  dom recs ( G )
65eleq2i 2380 . . . 4  |-  ( A  e.  dom  F  <->  A  e.  dom recs ( G ) )
74reseq1i 4988 . . . . 5  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87eleq1i 2379 . . . 4  |-  ( ( F  |`  A )  e.  _V  <->  (recs ( G )  |`  A )  e.  _V )
93, 6, 83bitr4g 279 . . 3  |-  ( A  e.  On  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
10 onprc 4613 . . . . . 6  |-  -.  On  e.  _V
11 elex 2830 . . . . . 6  |-  ( On  e.  dom  F  ->  On  e.  _V )
1210, 11mto 167 . . . . 5  |-  -.  On  e.  dom  F
13 eleq1 2376 . . . . 5  |-  ( A  =  On  ->  ( A  e.  dom  F  <->  On  e.  dom  F ) )
1412, 13mtbiri 294 . . . 4  |-  ( A  =  On  ->  -.  A  e.  dom  F )
152tfrlem13 6448 . . . . . 6  |-  -. recs ( G )  e.  _V
164eleq1i 2379 . . . . . 6  |-  ( F  e.  _V  <-> recs ( G
)  e.  _V )
1715, 16mtbir 290 . . . . 5  |-  -.  F  e.  _V
18 reseq2 4987 . . . . . . 7  |-  ( A  =  On  ->  ( F  |`  A )  =  ( F  |`  On ) )
194tfr1a 6452 . . . . . . . . . 10  |-  ( Fun 
F  /\  Lim  dom  F
)
2019simpli 444 . . . . . . . . 9  |-  Fun  F
21 funrel 5309 . . . . . . . . 9  |-  ( Fun 
F  ->  Rel  F )
2220, 21ax-mp 8 . . . . . . . 8  |-  Rel  F
2319simpri 448 . . . . . . . . 9  |-  Lim  dom  F
24 limord 4488 . . . . . . . . 9  |-  ( Lim 
dom  F  ->  Ord  dom  F )
25 ordsson 4618 . . . . . . . . 9  |-  ( Ord 
dom  F  ->  dom  F  C_  On )
2623, 24, 25mp2b 9 . . . . . . . 8  |-  dom  F  C_  On
27 relssres 5029 . . . . . . . 8  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2822, 26, 27mp2an 653 . . . . . . 7  |-  ( F  |`  On )  =  F
2918, 28syl6eq 2364 . . . . . 6  |-  ( A  =  On  ->  ( F  |`  A )  =  F )
3029eleq1d 2382 . . . . 5  |-  ( A  =  On  ->  (
( F  |`  A )  e.  _V  <->  F  e.  _V ) )
3117, 30mtbiri 294 . . . 4  |-  ( A  =  On  ->  -.  ( F  |`  A )  e.  _V )
3214, 312falsed 340 . . 3  |-  ( A  =  On  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
339, 32jaoi 368 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  e.  dom  F  <-> 
( F  |`  A )  e.  _V ) )
341, 33sylbi 187 1  |-  ( Ord 
A  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1633    e. wcel 1701   {cab 2302   A.wral 2577   E.wrex 2578   _Vcvv 2822    C_ wss 3186   Ord word 4428   Oncon0 4429   Lim wlim 4430   dom cdm 4726    |` cres 4728   Rel wrel 4731   Fun wfun 5286    Fn wfn 5287   ` cfv 5292  recscrecs 6429
This theorem is referenced by:  ordtypelem3  7280  ordtypelem9  7286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-recs 6430
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