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Theorem harwdom 7304
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 7259 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 2283 . . . . . 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 2283 . . . . . 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 7257 . . . . 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 965 . . . 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 964 . . . . 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 4800 . . . . . . 7  |-  ( ( X  e.  V  /\  X  e.  V )  ->  ( X  X.  X
)  e.  _V )
76anidms 626 . . . . . 6  |-  ( X  e.  V  ->  ( X  X.  X )  e. 
_V )
8 pwexg 4194 . . . . . 6  |-  ( ( X  X.  X )  e.  _V  ->  ~P ( X  X.  X
)  e.  _V )
97, 8syl 15 . . . . 5  |-  ( X  e.  V  ->  ~P ( X  X.  X
)  e.  _V )
10 ssexg 4160 . . . . 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 644 . . . 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 5743 . . . 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 644 . . 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 5283 . . . . . 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 199 . . . . 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 10 . . . 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 966 . . . . 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 7276 . . . . 5  |-  ( X  e.  V  ->  (har `  X )  =  {
x  e.  On  |  x  ~<_  X } )
1917, 18eqtr4d 2318 . . . 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 5261 . . . 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 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 ) ) } : 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 7285 . . 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 642 . 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 6907 . . . 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 1366 . . 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 7288 . . 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 15 . 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 7289 . 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 642 1  |-  ( X  e.  V  ->  (har `  X )  ~<_*  ~P ( X  X.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   {copab 4076    _E cep 4303    _I cid 4304    We wwe 4351   Oncon0 4392    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255    ~<_ cdom 6861  OrdIsocoi 7224  harchar 7270    ~<_* cwdom 7271
This theorem is referenced by:  gchhar  8293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-wdom 7273
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