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Theorem harwdom 8015
Description: The Hartogs function is weakly dominated by  ~P ( X  X.  X ). This follows from a more precise analysis of the bound used in hartogs 7968 to prove that  (har `  X ) is a set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harwdom  |-  ( X  e.  V  ->  (har `  X )  ~<_*  ~P ( X  X.  X ) )

Proof of Theorem harwdom
Dummy variables  y 
r  f  s  t  w  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . 6  |-  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  =  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }
2 eqid 2467 . . . . . 6  |-  { <. s ,  t >.  |  E. w  e.  y  E. z  e.  y  (
( s  =  ( f `  w )  /\  t  =  ( f `  z ) )  /\  w  _E  z ) }  =  { <. s ,  t
>.  |  E. w  e.  y  E. z  e.  y  ( (
s  =  ( f `
 w )  /\  t  =  ( f `  z ) )  /\  w  _E  z ) }
31, 2hartogslem1 7966 . . . . 5  |-  ( dom 
{ <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  C_  ~P ( X  X.  X )  /\  Fun  { <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  /\  ( X  e.  V  ->  ran  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  =  { x  e.  On  |  x  ~<_  X } ) )
43simp2i 1006 . . . 4  |-  Fun  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }
53simp1i 1005 . . . . 5  |-  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  C_  ~P ( X  X.  X )
6 xpexg 6710 . . . . . . 7  |-  ( ( X  e.  V  /\  X  e.  V )  ->  ( X  X.  X
)  e.  _V )
76anidms 645 . . . . . 6  |-  ( X  e.  V  ->  ( X  X.  X )  e. 
_V )
8 pwexg 4631 . . . . . 6  |-  ( ( X  X.  X )  e.  _V  ->  ~P ( X  X.  X
)  e.  _V )
97, 8syl 16 . . . . 5  |-  ( X  e.  V  ->  ~P ( X  X.  X
)  e.  _V )
10 ssexg 4593 . . . . 5  |-  ( ( dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  C_  ~P ( X  X.  X )  /\  ~P ( X  X.  X
)  e.  _V )  ->  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  e.  _V )
115, 9, 10sylancr 663 . . . 4  |-  ( X  e.  V  ->  dom  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  e.  _V )
12 funex 6127 . . . 4  |-  ( ( Fun  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  /\  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  e.  _V )  ->  { <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  e.  _V )
134, 11, 12sylancr 663 . . 3  |-  ( X  e.  V  ->  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  e.  _V )
14 funfn 5616 . . . . . 6  |-  ( Fun 
{ <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  <->  { <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  Fn  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) } )
154, 14mpbi 208 . . . . 5  |-  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  Fn  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }
1615a1i 11 . . . 4  |-  ( X  e.  V  ->  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  Fn  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) } )
173simp3i 1007 . . . . 5  |-  ( X  e.  V  ->  ran  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  =  { x  e.  On  |  x  ~<_  X } )
18 harval 7987 . . . . 5  |-  ( X  e.  V  ->  (har `  X )  =  {
x  e.  On  |  x  ~<_  X } )
1917, 18eqtr4d 2511 . . . 4  |-  ( X  e.  V  ->  ran  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  =  (har `  X ) )
20 df-fo 5593 . . . 4  |-  ( {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) } : dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) } -onto-> (har `  X )  <->  ( { <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  Fn  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  /\  ran  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  =  (har `  X ) ) )
2116, 19, 20sylanbrc 664 . . 3  |-  ( X  e.  V  ->  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) } : dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) } -onto-> (har `  X )
)
22 fowdom 7996 . . 3  |-  ( ( { <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  e.  _V  /\  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) } : dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) } -onto-> (har `  X )
)  ->  (har `  X
)  ~<_*  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) } )
2313, 21, 22syl2anc 661 . 2  |-  ( X  e.  V  ->  (har `  X )  ~<_*  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) } )
24 ssdomg 7561 . . . 4  |-  ( ~P ( X  X.  X
)  e.  _V  ->  ( dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  C_  ~P ( X  X.  X )  ->  dom  { <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  ~<_  ~P ( X  X.  X ) ) )
259, 5, 24mpisyl 18 . . 3  |-  ( X  e.  V  ->  dom  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  ~<_  ~P ( X  X.  X ) )
26 domwdom 7999 . . 3  |-  ( dom 
{ <. r ,  y
>.  |  ( (
( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  ~<_  ~P ( X  X.  X )  ->  dom  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  ~<_*  ~P ( X  X.  X ) )
2725, 26syl 16 . 2  |-  ( X  e.  V  ->  dom  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  ~<_*  ~P ( X  X.  X ) )
28 wdomtr 8000 . 2  |-  ( ( (har `  X )  ~<_*  dom  {
<. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  /\  dom  { <. r ,  y >.  |  ( ( ( dom  r  C_  X  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }  ~<_*  ~P ( X  X.  X ) )  -> 
(har `  X )  ~<_*  ~P ( X  X.  X
) )
2923, 27, 28syl2anc 661 1  |-  ( X  e.  V  ->  (har `  X )  ~<_*  ~P ( X  X.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113    \ cdif 3473    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   {copab 4504    _E cep 4789    _I cid 4790    We wwe 4837   Oncon0 4878    X. cxp 4997   dom cdm 4999   ran crn 5000    |` cres 5001   Fun wfun 5581    Fn wfn 5582   -onto->wfo 5585   ` cfv 5587    ~<_ cdom 7514  OrdIsocoi 7933  harchar 7981    ~<_* cwdom 7982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-recs 7042  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-oi 7934  df-har 7983  df-wdom 7984
This theorem is referenced by:  gchhar  9056
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