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Theorem tfr2b 7062
Description: Without assuming ax-rep 4558, 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 6602 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 eqid 2467 . . . . 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 7058 . . . 4  |-  ( A  e.  On  ->  ( A  e.  dom recs ( G )  <->  (recs ( G )  |`  A )  e.  _V ) )
4 tfr.1 . . . . . 6  |-  F  = recs ( G )
54dmeqi 5202 . . . . 5  |-  dom  F  =  dom recs ( G )
65eleq2i 2545 . . . 4  |-  ( A  e.  dom  F  <->  A  e.  dom recs ( G ) )
74reseq1i 5267 . . . . 5  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87eleq1i 2544 . . . 4  |-  ( ( F  |`  A )  e.  _V  <->  (recs ( G )  |`  A )  e.  _V )
93, 6, 83bitr4g 288 . . 3  |-  ( A  e.  On  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
10 onprc 6598 . . . . . 6  |-  -.  On  e.  _V
11 elex 3122 . . . . . 6  |-  ( On  e.  dom  F  ->  On  e.  _V )
1210, 11mto 176 . . . . 5  |-  -.  On  e.  dom  F
13 eleq1 2539 . . . . 5  |-  ( A  =  On  ->  ( A  e.  dom  F  <->  On  e.  dom  F ) )
1412, 13mtbiri 303 . . . 4  |-  ( A  =  On  ->  -.  A  e.  dom  F )
152tfrlem13 7056 . . . . . 6  |-  -. recs ( G )  e.  _V
164eleq1i 2544 . . . . . 6  |-  ( F  e.  _V  <-> recs ( G
)  e.  _V )
1715, 16mtbir 299 . . . . 5  |-  -.  F  e.  _V
18 reseq2 5266 . . . . . . 7  |-  ( A  =  On  ->  ( F  |`  A )  =  ( F  |`  On ) )
194tfr1a 7060 . . . . . . . . . 10  |-  ( Fun 
F  /\  Lim  dom  F
)
2019simpli 458 . . . . . . . . 9  |-  Fun  F
21 funrel 5603 . . . . . . . . 9  |-  ( Fun 
F  ->  Rel  F )
2220, 21ax-mp 5 . . . . . . . 8  |-  Rel  F
2319simpri 462 . . . . . . . . 9  |-  Lim  dom  F
24 limord 4937 . . . . . . . . 9  |-  ( Lim 
dom  F  ->  Ord  dom  F )
25 ordsson 6603 . . . . . . . . 9  |-  ( Ord 
dom  F  ->  dom  F  C_  On )
2623, 24, 25mp2b 10 . . . . . . . 8  |-  dom  F  C_  On
27 relssres 5309 . . . . . . . 8  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2822, 26, 27mp2an 672 . . . . . . 7  |-  ( F  |`  On )  =  F
2918, 28syl6eq 2524 . . . . . 6  |-  ( A  =  On  ->  ( F  |`  A )  =  F )
3029eleq1d 2536 . . . . 5  |-  ( A  =  On  ->  (
( F  |`  A )  e.  _V  <->  F  e.  _V ) )
3117, 30mtbiri 303 . . . 4  |-  ( A  =  On  ->  -.  ( F  |`  A )  e.  _V )
3214, 312falsed 351 . . 3  |-  ( A  =  On  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
339, 32jaoi 379 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  e.  dom  F  <-> 
( F  |`  A )  e.  _V ) )
341, 33sylbi 195 1  |-  ( Ord 
A  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   Ord word 4877   Oncon0 4878   Lim wlim 4879   dom cdm 4999    |` cres 5001   Rel wrel 5004   Fun wfun 5580    Fn wfn 5581   ` cfv 5586  recscrecs 7038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-recs 7039
This theorem is referenced by:  ordtypelem3  7941  ordtypelem9  7947
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