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Theorem hsmex 8148
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 7396. (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 4108 . . . . 5  |-  ( a  =  X  ->  (
x  ~<_  a  <->  x  ~<_  X ) )
21ralbidv 2639 . . . 4  |-  ( a  =  X  ->  ( A. x  e.  ( TC `  { s } ) x  ~<_  a  <->  A. x  e.  ( TC `  {
s } ) x  ~<_  X ) )
32rabbidv 2856 . . 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 2424 . 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 2867 . . 3  |-  a  e. 
_V
6 eqid 2358 . . 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 6512 . . . . . 6  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( f  e. 
_V  |->  U. f ) ,  b ) )
8 unieq 3917 . . . . . . . 8  |-  ( f  =  c  ->  U. f  =  U. c )
98cbvmptv 4192 . . . . . . 7  |-  ( f  e.  _V  |->  U. f
)  =  ( c  e.  _V  |->  U. c
)
10 rdgeq1 6511 . . . . . . 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 2406 . . . . 5  |-  ( e  =  b  ->  rec ( ( f  e. 
_V  |->  U. f ) ,  e )  =  rec ( ( c  e. 
_V  |->  U. c ) ,  b ) )
1312reseq1d 5036 . . . 4  |-  ( e  =  b  ->  ( rec ( ( f  e. 
_V  |->  U. f ) ,  e )  |`  om )  =  ( rec (
( c  e.  _V  |->  U. c ) ,  b )  |`  om )
)
1413cbvmptv 4192 . . 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 2358 . . 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 2358 . . 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 8147 . 2  |-  { s  e.  U. ( R1
" On )  | 
A. x  e.  ( TC `  { s } ) x  ~<_  a }  e.  _V
184, 17vtoclg 2919 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 1642    e. wcel 1710   A.wral 2619   {crab 2623   _Vcvv 2864   ~Pcpw 3701   {csn 3716   U.cuni 3908   class class class wbr 4104    e. cmpt 4158    _E cep 4385   Oncon0 4474   omcom 4738    X. cxp 4769    |` cres 4773   "cima 4774   ` cfv 5337   reccrdg 6509    ~<_ cdom 6949  OrdIsocoi 7314  harchar 7360   TCctc 7511   R1cr1 7524   rankcrnk 7525
This theorem is referenced by:  hsmex2  8149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-1st 6209  df-2nd 6210  df-riota 6391  df-smo 6450  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-oi 7315  df-har 7362  df-wdom 7363  df-tc 7512  df-r1 7526  df-rank 7527
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