MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfr2b Unicode version

Theorem tfr2b 6412
Description: Without assuming ax-rep 4131, 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 4580 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 eqid 2283 . . . . 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 6408 . . . 4  |-  ( A  e.  On  ->  ( A  e.  dom recs ( G )  <->  (recs ( G )  |`  A )  e.  _V ) )
4 tfr.1 . . . . . 6  |-  F  = recs ( G )
54dmeqi 4880 . . . . 5  |-  dom  F  =  dom recs ( G )
65eleq2i 2347 . . . 4  |-  ( A  e.  dom  F  <->  A  e.  dom recs ( G ) )
74reseq1i 4951 . . . . 5  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87eleq1i 2346 . . . 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 4576 . . . . . 6  |-  -.  On  e.  _V
11 elex 2796 . . . . . 6  |-  ( On  e.  dom  F  ->  On  e.  _V )
1210, 11mto 167 . . . . 5  |-  -.  On  e.  dom  F
13 eleq1 2343 . . . . 5  |-  ( A  =  On  ->  ( A  e.  dom  F  <->  On  e.  dom  F ) )
1412, 13mtbiri 294 . . . 4  |-  ( A  =  On  ->  -.  A  e.  dom  F )
152tfrlem13 6406 . . . . . 6  |-  -. recs ( G )  e.  _V
164eleq1i 2346 . . . . . 6  |-  ( F  e.  _V  <-> recs ( G
)  e.  _V )
1715, 16mtbir 290 . . . . 5  |-  -.  F  e.  _V
18 reseq2 4950 . . . . . . 7  |-  ( A  =  On  ->  ( F  |`  A )  =  ( F  |`  On ) )
194tfr1a 6410 . . . . . . . . . 10  |-  ( Fun 
F  /\  Lim  dom  F
)
2019simpli 444 . . . . . . . . 9  |-  Fun  F
21 funrel 5272 . . . . . . . . 9  |-  ( Fun 
F  ->  Rel  F )
2220, 21ax-mp 8 . . . . . . . 8  |-  Rel  F
2319simpri 448 . . . . . . . . 9  |-  Lim  dom  F
24 limord 4451 . . . . . . . . 9  |-  ( Lim 
dom  F  ->  Ord  dom  F )
25 ordsson 4581 . . . . . . . . 9  |-  ( Ord 
dom  F  ->  dom  F  C_  On )
2623, 24, 25mp2b 9 . . . . . . . 8  |-  dom  F  C_  On
27 relssres 4992 . . . . . . . 8  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2822, 26, 27mp2an 653 . . . . . . 7  |-  ( F  |`  On )  =  F
2918, 28syl6eq 2331 . . . . . 6  |-  ( A  =  On  ->  ( F  |`  A )  =  F )
3029eleq1d 2349 . . . . 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 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   Ord word 4391   Oncon0 4392   Lim wlim 4393   dom cdm 4689    |` cres 4691   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  ordtypelem3  7235  ordtypelem9  7241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388
  Copyright terms: Public domain W3C validator