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Theorem hsmex 8801
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 8007. (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 4444 . . . . 5  |-  ( a  =  X  ->  (
x  ~<_  a  <->  x  ~<_  X ) )
21ralbidv 2896 . . . 4  |-  ( a  =  X  ->  ( A. x  e.  ( TC `  { s } ) x  ~<_  a  <->  A. x  e.  ( TC `  {
s } ) x  ~<_  X ) )
32rabbidv 3098 . . 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 2529 . 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 3109 . . 3  |-  a  e. 
_V
6 eqid 2460 . . 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 7068 . . . . . 6  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( f  e. 
_V  |->  U. f ) ,  b ) )
8 unieq 4246 . . . . . . . 8  |-  ( f  =  c  ->  U. f  =  U. c )
98cbvmptv 4531 . . . . . . 7  |-  ( f  e.  _V  |->  U. f
)  =  ( c  e.  _V  |->  U. c
)
10 rdgeq1 7067 . . . . . . 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 2517 . . . . 5  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( c  e. 
_V  |->  U. c ) ,  b ) )
1312reseq1d 5263 . . . 4  |-  ( e  =  b  ->  ( rec ( ( f  e. 
_V  |->  U. f ) ,  e )  |`  om )  =  ( rec (
( c  e.  _V  |->  U. c ) ,  b )  |`  om )
)
1413cbvmptv 4531 . . 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 2460 . . 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 2460 . . 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 8800 . 2  |-  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  a }  e.  _V
184, 17vtoclg 3164 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 1374    e. wcel 1762   A.wral 2807   {crab 2811   _Vcvv 3106   ~Pcpw 4003   {csn 4020   U.cuni 4238   class class class wbr 4440    |-> cmpt 4498    _E cep 4782   Oncon0 4871    X. cxp 4990    |` cres 4994   "cima 4995   ` cfv 5579   omcom 6671   reccrdg 7065    ~<_ cdom 7504  OrdIsocoi 7923  harchar 7971   TCctc 8156   R1cr1 8169   rankcrnk 8170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-om 6672  df-1st 6774  df-2nd 6775  df-smo 7007  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-oi 7924  df-har 7973  df-wdom 7974  df-tc 8157  df-r1 8171  df-rank 8172
This theorem is referenced by:  hsmex2  8802
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