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Theorem tfr2b 7083
Description: Without assuming ax-rep 4568, 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 6623 . 2  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
2 eqid 2457 . . . . 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 7079 . . . 4  |-  ( A  e.  On  ->  ( A  e.  dom recs ( G )  <->  (recs ( G )  |`  A )  e.  _V ) )
4 tfr.1 . . . . . 6  |-  F  = recs ( G )
54dmeqi 5214 . . . . 5  |-  dom  F  =  dom recs ( G )
65eleq2i 2535 . . . 4  |-  ( A  e.  dom  F  <->  A  e.  dom recs ( G ) )
74reseq1i 5279 . . . . 5  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87eleq1i 2534 . . . 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 6619 . . . . . 6  |-  -.  On  e.  _V
11 elex 3118 . . . . . 6  |-  ( On  e.  dom  F  ->  On  e.  _V )
1210, 11mto 176 . . . . 5  |-  -.  On  e.  dom  F
13 eleq1 2529 . . . . 5  |-  ( A  =  On  ->  ( A  e.  dom  F  <->  On  e.  dom  F ) )
1412, 13mtbiri 303 . . . 4  |-  ( A  =  On  ->  -.  A  e.  dom  F )
152tfrlem13 7077 . . . . . 6  |-  -. recs ( G )  e.  _V
164eleq1i 2534 . . . . . 6  |-  ( F  e.  _V  <-> recs ( G
)  e.  _V )
1715, 16mtbir 299 . . . . 5  |-  -.  F  e.  _V
18 reseq2 5278 . . . . . . 7  |-  ( A  =  On  ->  ( F  |`  A )  =  ( F  |`  On ) )
194tfr1a 7081 . . . . . . . . . 10  |-  ( Fun 
F  /\  Lim  dom  F
)
2019simpli 458 . . . . . . . . 9  |-  Fun  F
21 funrel 5611 . . . . . . . . 9  |-  ( Fun 
F  ->  Rel  F )
2220, 21ax-mp 5 . . . . . . . 8  |-  Rel  F
2319simpri 462 . . . . . . . . 9  |-  Lim  dom  F
24 limord 4946 . . . . . . . . 9  |-  ( Lim 
dom  F  ->  Ord  dom  F )
25 ordsson 6624 . . . . . . . . 9  |-  ( Ord 
dom  F  ->  dom  F  C_  On )
2623, 24, 25mp2b 10 . . . . . . . 8  |-  dom  F  C_  On
27 relssres 5321 . . . . . . . 8  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2822, 26, 27mp2an 672 . . . . . . 7  |-  ( F  |`  On )  =  F
2918, 28syl6eq 2514 . . . . . 6  |-  ( A  =  On  ->  ( F  |`  A )  =  F )
3029eleq1d 2526 . . . . 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 1395    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808   _Vcvv 3109    C_ wss 3471   Ord word 4886   Oncon0 4887   Lim wlim 4888   dom cdm 5008    |` cres 5010   Rel wrel 5013   Fun wfun 5588    Fn wfn 5589   ` cfv 5594  recscrecs 7059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-recs 7060
This theorem is referenced by:  ordtypelem3  7963  ordtypelem9  7969
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