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Theorem harwdom 7320
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 7275 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 2296 . . . . . 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 2296 . . . . . 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 7273 . . . . 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 4816 . . . . . . 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 4210 . . . . . 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 4176 . . . . 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 5759 . . . 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 5299 . . . . . 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 7292 . . . . 5  |-  ( X  e.  V  ->  (har `  X )  =  {
x  e.  On  |  x  ~<_  X } )
1917, 18eqtr4d 2331 . . . 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 5277 . . . 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 7301 . . 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 6923 . . . 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 7304 . . 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 7305 . 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 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   {copab 4092    _E cep 4319    _I cid 4320    We wwe 4367   Oncon0 4408    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707   Fun wfun 5265    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271    ~<_ cdom 6877  OrdIsocoi 7240  harchar 7286    ~<_* cwdom 7287
This theorem is referenced by:  gchhar  8309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-oi 7241  df-har 7288  df-wdom 7289
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