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Theorem harwdom 8105
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 8059 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 2429 . . . . . 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 2429 . . . . . 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 8057 . . . . 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 1015 . . . 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 1014 . . . . 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 sqxpexg 6610 . . . . . 6  |-  ( X  e.  V  ->  ( X  X.  X )  e. 
_V )
7 pwexg 4609 . . . . . 6  |-  ( ( X  X.  X )  e.  _V  ->  ~P ( X  X.  X
)  e.  _V )
86, 7syl 17 . . . . 5  |-  ( X  e.  V  ->  ~P ( X  X.  X
)  e.  _V )
9 ssexg 4571 . . . . 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 )
105, 8, 9sylancr 667 . . . 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 )
11 funex 6148 . . . 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 )
124, 10, 11sylancr 667 . . 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 )
13 funfn 5630 . . . . . 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 ) ) } )
144, 13mpbi 211 . . . . 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 ) ) }
1514a1i 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 ) ) } )
163simp3i 1016 . . . . 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 } )
17 harval 8077 . . . . 5  |-  ( X  e.  V  ->  (har `  X )  =  {
x  e.  On  |  x  ~<_  X } )
1816, 17eqtr4d 2473 . . . 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 ) )
19 df-fo 5607 . . . 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 ) ) )
2015, 18, 19sylanbrc 668 . . 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 )
)
21 fowdom 8086 . . 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 ) ) } )
2212, 20, 21syl2anc 665 . 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 ) ) } )
23 ssdomg 7622 . . . 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 ) ) )
248, 5, 23mpisyl 22 . . 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 ) )
25 domwdom 8089 . . 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 ) )
2624, 25syl 17 . 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 ) )
27 wdomtr 8090 . 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
) )
2822, 26, 27syl2anc 665 1  |-  ( X  e.  V  ->  (har `  X )  ~<_*  ~P ( X  X.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   E.wrex 2783   {crab 2786   _Vcvv 3087    \ cdif 3439    C_ wss 3442   ~Pcpw 3985   class class class wbr 4426   {copab 4483    _E cep 4763    _I cid 4764    We wwe 4812    X. cxp 4852   dom cdm 4854   ran crn 4855    |` cres 4856   Oncon0 5442   Fun wfun 5595    Fn wfn 5596   -onto->wfo 5599   ` cfv 5601    ~<_ cdom 7575  OrdIsocoi 8024  harchar 8071    ~<_* cwdom 8072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-wrecs 7036  df-recs 7098  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-oi 8025  df-har 8073  df-wdom 8074
This theorem is referenced by:  gchhar  9103
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