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Theorem hsmex 8058
Description: The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 7306. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Assertion
Ref Expression
hsmex  |-  ( X  e.  V  ->  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  X }  e.  _V )
Distinct variable group:    x, s, X
Allowed substitution hints:    V( x, s)

Proof of Theorem hsmex
Dummy variables  a 
b  c  d  e  f  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4027 . . . . 5  |-  ( a  =  X  ->  (
x  ~<_  a  <->  x  ~<_  X ) )
21ralbidv 2563 . . . 4  |-  ( a  =  X  ->  ( A. x  e.  ( TC `  { s } ) x  ~<_  a  <->  A. x  e.  ( TC `  {
s } ) x  ~<_  X ) )
32rabbidv 2780 . . 3  |-  ( a  =  X  ->  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  a }  =  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  X } )
43eleq1d 2349 . 2  |-  ( a  =  X  ->  ( { s  e.  U. ( R1 " On )  |  A. x  e.  ( TC `  {
s } ) x  ~<_  a }  e.  _V  <->  { s  e.  U. ( R1 " On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  X }  e.  _V )
)
5 vex 2791 . . 3  |-  a  e. 
_V
6 eqid 2283 . . 3  |-  ( rec ( ( d  e. 
_V  |->  (har `  ~P ( a  X.  d
) ) ) ,  (har `  ~P a
) )  |`  om )  =  ( rec (
( d  e.  _V  |->  (har `  ~P ( a  X.  d ) ) ) ,  (har `  ~P a ) )  |`  om )
7 rdgeq2 6425 . . . . . 6  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( f  e. 
_V  |->  U. f ) ,  b ) )
8 unieq 3836 . . . . . . . 8  |-  ( f  =  c  ->  U. f  =  U. c )
98cbvmptv 4111 . . . . . . 7  |-  ( f  e.  _V  |->  U. f
)  =  ( c  e.  _V  |->  U. c
)
10 rdgeq1 6424 . . . . . . 7  |-  ( ( f  e.  _V  |->  U. f )  =  ( c  e.  _V  |->  U. c )  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  b )  =  rec ( ( c  e. 
_V  |->  U. c ) ,  b ) )
119, 10ax-mp 8 . . . . . 6  |-  rec (
( f  e.  _V  |->  U. f ) ,  b )  =  rec (
( c  e.  _V  |->  U. c ) ,  b )
127, 11syl6eq 2331 . . . . 5  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( c  e. 
_V  |->  U. c ) ,  b ) )
1312reseq1d 4954 . . . 4  |-  ( e  =  b  ->  ( rec ( ( f  e. 
_V  |->  U. f ) ,  e )  |`  om )  =  ( rec (
( c  e.  _V  |->  U. c ) ,  b )  |`  om )
)
1413cbvmptv 4111 . . 3  |-  ( e  e.  _V  |->  ( rec ( ( f  e. 
_V  |->  U. f ) ,  e )  |`  om )
)  =  ( b  e.  _V  |->  ( rec ( ( c  e. 
_V  |->  U. c ) ,  b )  |`  om )
)
15 eqid 2283 . . 3  |-  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  a }  =  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  a }
16 eqid 2283 . . 3  |- OrdIso (  _E  ,  ( rank " (
( ( e  e. 
_V  |->  ( rec (
( f  e.  _V  |->  U. f ) ,  e )  |`  om )
) `  z ) `  y ) ) )  = OrdIso (  _E  , 
( rank " ( ( ( e  e.  _V  |->  ( rec ( ( f  e.  _V  |->  U. f
) ,  e )  |`  om ) ) `  z ) `  y
) ) )
175, 6, 14, 15, 16hsmexlem6 8057 . 2  |-  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  a }  e.  _V
184, 17vtoclg 2843 1  |-  ( X  e.  V  ->  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  X }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    _E cep 4303   Oncon0 4392   omcom 4656    X. cxp 4687    |` cres 4691   "cima 4692   ` cfv 5255   reccrdg 6422    ~<_ cdom 6861  OrdIsocoi 7224  harchar 7270   TCctc 7421   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  hsmex2  8059
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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-rmo 2551  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-int 3863  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-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-isom 5264  df-1st 6122  df-2nd 6123  df-riota 6304  df-smo 6363  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-wdom 7273  df-tc 7422  df-r1 7436  df-rank 7437
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