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Theorem hsmex 8272
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 7520. (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 4180 . . . . 5  |-  ( a  =  X  ->  (
x  ~<_  a  <->  x  ~<_  X ) )
21ralbidv 2690 . . . 4  |-  ( a  =  X  ->  ( A. x  e.  ( TC `  { s } ) x  ~<_  a  <->  A. x  e.  ( TC `  {
s } ) x  ~<_  X ) )
32rabbidv 2912 . . 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 2474 . 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 2923 . . 3  |-  a  e. 
_V
6 eqid 2408 . . 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 6633 . . . . . 6  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( f  e. 
_V  |->  U. f ) ,  b ) )
8 unieq 3988 . . . . . . . 8  |-  ( f  =  c  ->  U. f  =  U. c )
98cbvmptv 4264 . . . . . . 7  |-  ( f  e.  _V  |->  U. f
)  =  ( c  e.  _V  |->  U. c
)
10 rdgeq1 6632 . . . . . . 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 2456 . . . . 5  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( c  e. 
_V  |->  U. c ) ,  b ) )
1312reseq1d 5108 . . . 4  |-  ( e  =  b  ->  ( rec ( ( f  e. 
_V  |->  U. f ) ,  e )  |`  om )  =  ( rec (
( c  e.  _V  |->  U. c ) ,  b )  |`  om )
)
1413cbvmptv 4264 . . 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 2408 . . 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 2408 . . 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 8271 . 2  |-  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  a }  e.  _V
184, 17vtoclg 2975 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 1649    e. wcel 1721   A.wral 2670   {crab 2674   _Vcvv 2920   ~Pcpw 3763   {csn 3778   U.cuni 3979   class class class wbr 4176    e. cmpt 4230    _E cep 4456   Oncon0 4545   omcom 4808    X. cxp 4839    |` cres 4843   "cima 4844   ` cfv 5417   reccrdg 6630    ~<_ cdom 7070  OrdIsocoi 7438  harchar 7484   TCctc 7635   R1cr1 7648   rankcrnk 7649
This theorem is referenced by:  hsmex2  8273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-1st 6312  df-2nd 6313  df-riota 6512  df-smo 6571  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-oi 7439  df-har 7486  df-wdom 7487  df-tc 7636  df-r1 7650  df-rank 7651
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