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Theorem harwdom 7491
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 7446 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 2387 . . . . . 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 2387 . . . . . 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 7444 . . . . 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 967 . . . 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 966 . . . . 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 4929 . . . . . . 7  |-  ( ( X  e.  V  /\  X  e.  V )  ->  ( X  X.  X
)  e.  _V )
76anidms 627 . . . . . 6  |-  ( X  e.  V  ->  ( X  X.  X )  e. 
_V )
8 pwexg 4324 . . . . . 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 4290 . . . . 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 645 . . . 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 5902 . . . 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 645 . . 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 5422 . . . . . 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 200 . . . . 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 968 . . . . 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 7463 . . . . 5  |-  ( X  e.  V  ->  (har `  X )  =  {
x  e.  On  |  x  ~<_  X } )
1917, 18eqtr4d 2422 . . . 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 5400 . . . 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 646 . . 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 7472 . . 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 643 . 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 7089 . . . 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, 24ee10 1382 . . 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 7475 . . 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 7476 . 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 643 1  |-  ( X  e.  V  ->  (har `  X )  ~<_*  ~P ( X  X.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2650   {crab 2653   _Vcvv 2899    \ cdif 3260    C_ wss 3263   ~Pcpw 3742   class class class wbr 4153   {copab 4206    _E cep 4433    _I cid 4434    We wwe 4481   Oncon0 4522    X. cxp 4816   dom cdm 4818   ran crn 4819    |` cres 4820   Fun wfun 5388    Fn wfn 5389   -onto->wfo 5392   ` cfv 5394    ~<_ cdom 7043  OrdIsocoi 7411  harchar 7457    ~<_* cwdom 7458
This theorem is referenced by:  gchhar  8479
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-riota 6485  df-recs 6569  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-oi 7412  df-har 7459  df-wdom 7460
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