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Theorem hsmex 8713
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 7919. (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 4405 . . . . 5  |-  ( a  =  X  ->  (
x  ~<_  a  <->  x  ~<_  X ) )
21ralbidv 2846 . . . 4  |-  ( a  =  X  ->  ( A. x  e.  ( TC `  { s } ) x  ~<_  a  <->  A. x  e.  ( TC `  {
s } ) x  ~<_  X ) )
32rabbidv 3070 . . 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 2523 . 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 3081 . . 3  |-  a  e. 
_V
6 eqid 2454 . . 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 6979 . . . . . 6  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( f  e. 
_V  |->  U. f ) ,  b ) )
8 unieq 4208 . . . . . . . 8  |-  ( f  =  c  ->  U. f  =  U. c )
98cbvmptv 4492 . . . . . . 7  |-  ( f  e.  _V  |->  U. f
)  =  ( c  e.  _V  |->  U. c
)
10 rdgeq1 6978 . . . . . . 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 5 . . . . . 6  |-  rec (
( f  e.  _V  |->  U. f ) ,  b )  =  rec (
( c  e.  _V  |->  U. c ) ,  b )
127, 11syl6eq 2511 . . . . 5  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( c  e. 
_V  |->  U. c ) ,  b ) )
1312reseq1d 5218 . . . 4  |-  ( e  =  b  ->  ( rec ( ( f  e. 
_V  |->  U. f ) ,  e )  |`  om )  =  ( rec (
( c  e.  _V  |->  U. c ) ,  b )  |`  om )
)
1413cbvmptv 4492 . . 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 2454 . . 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 2454 . . 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 8712 . 2  |-  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  a }  e.  _V
184, 17vtoclg 3136 1  |-  ( X  e.  V  ->  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  X }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   _Vcvv 3078   ~Pcpw 3969   {csn 3986   U.cuni 4200   class class class wbr 4401    |-> cmpt 4459    _E cep 4739   Oncon0 4828    X. cxp 4947    |` cres 4951   "cima 4952   ` cfv 5527   omcom 6587   reccrdg 6976    ~<_ cdom 7419  OrdIsocoi 7835  harchar 7883   TCctc 8068   R1cr1 8081   rankcrnk 8082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-om 6588  df-1st 6688  df-2nd 6689  df-smo 6918  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-oi 7836  df-har 7885  df-wdom 7886  df-tc 8069  df-r1 8083  df-rank 8084
This theorem is referenced by:  hsmex2  8714
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