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Theorem tfr2b 6659
Description: Without assuming ax-rep 4322, 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 4771 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 eqid 2438 . . . . 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 6655 . . . 4  |-  ( A  e.  On  ->  ( A  e.  dom recs ( G )  <->  (recs ( G )  |`  A )  e.  _V ) )
4 tfr.1 . . . . . 6  |-  F  = recs ( G )
54dmeqi 5073 . . . . 5  |-  dom  F  =  dom recs ( G )
65eleq2i 2502 . . . 4  |-  ( A  e.  dom  F  <->  A  e.  dom recs ( G ) )
74reseq1i 5144 . . . . 5  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87eleq1i 2501 . . . 4  |-  ( ( F  |`  A )  e.  _V  <->  (recs ( G )  |`  A )  e.  _V )
93, 6, 83bitr4g 281 . . 3  |-  ( A  e.  On  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
10 onprc 4767 . . . . . 6  |-  -.  On  e.  _V
11 elex 2966 . . . . . 6  |-  ( On  e.  dom  F  ->  On  e.  _V )
1210, 11mto 170 . . . . 5  |-  -.  On  e.  dom  F
13 eleq1 2498 . . . . 5  |-  ( A  =  On  ->  ( A  e.  dom  F  <->  On  e.  dom  F ) )
1412, 13mtbiri 296 . . . 4  |-  ( A  =  On  ->  -.  A  e.  dom  F )
152tfrlem13 6653 . . . . . 6  |-  -. recs ( G )  e.  _V
164eleq1i 2501 . . . . . 6  |-  ( F  e.  _V  <-> recs ( G
)  e.  _V )
1715, 16mtbir 292 . . . . 5  |-  -.  F  e.  _V
18 reseq2 5143 . . . . . . 7  |-  ( A  =  On  ->  ( F  |`  A )  =  ( F  |`  On ) )
194tfr1a 6657 . . . . . . . . . 10  |-  ( Fun 
F  /\  Lim  dom  F
)
2019simpli 446 . . . . . . . . 9  |-  Fun  F
21 funrel 5473 . . . . . . . . 9  |-  ( Fun 
F  ->  Rel  F )
2220, 21ax-mp 8 . . . . . . . 8  |-  Rel  F
2319simpri 450 . . . . . . . . 9  |-  Lim  dom  F
24 limord 4642 . . . . . . . . 9  |-  ( Lim 
dom  F  ->  Ord  dom  F )
25 ordsson 4772 . . . . . . . . 9  |-  ( Ord 
dom  F  ->  dom  F  C_  On )
2623, 24, 25mp2b 10 . . . . . . . 8  |-  dom  F  C_  On
27 relssres 5185 . . . . . . . 8  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2822, 26, 27mp2an 655 . . . . . . 7  |-  ( F  |`  On )  =  F
2918, 28syl6eq 2486 . . . . . 6  |-  ( A  =  On  ->  ( F  |`  A )  =  F )
3029eleq1d 2504 . . . . 5  |-  ( A  =  On  ->  (
( F  |`  A )  e.  _V  <->  F  e.  _V ) )
3117, 30mtbiri 296 . . . 4  |-  ( A  =  On  ->  -.  ( F  |`  A )  e.  _V )
3214, 312falsed 342 . . 3  |-  ( A  =  On  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
339, 32jaoi 370 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  e.  dom  F  <-> 
( F  |`  A )  e.  _V ) )
341, 33sylbi 189 1  |-  ( Ord 
A  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   Ord word 4582   Oncon0 4583   Lim wlim 4584   dom cdm 4880    |` cres 4882   Rel wrel 4885   Fun wfun 5450    Fn wfn 5451   ` cfv 5456  recscrecs 6634
This theorem is referenced by:  ordtypelem3  7491  ordtypelem9  7497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635
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