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Theorem hartogslem2 7969
Description: Lemma for hartogs 7970. (Contributed by Mario Carneiro, 14-Jan-2013.)
Hypotheses
Ref Expression
hartogslem.2  |-  F  =  { <. r ,  y
>.  |  ( (
( dom  r  C_  A  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r ) ) }
hartogslem.3  |-  R  =  { <. s ,  t
>.  |  E. w  e.  y  E. z  e.  y  ( (
s  =  ( f `
 w )  /\  t  =  ( f `  z ) )  /\  w  _E  z ) }
Assertion
Ref Expression
hartogslem2  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
Distinct variable groups:    f, s,
t, w, y, z   
f, r, x, A, y    R, r, x    V, r, y
Allowed substitution hints:    A( z, w, t, s)    R( y, z, w, t, f, s)    F( x, y, z, w, t, f, s, r)    V( x, z, w, t, f, s)

Proof of Theorem hartogslem2
StepHypRef Expression
1 hartogslem.2 . . . 4  |-  F  =  { <. r ,  y
>.  |  ( (
( dom  r  C_  A  /\  (  _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 hartogslem.3 . . . 4  |-  R  =  { <. s ,  t
>.  |  E. w  e.  y  E. z  e.  y  ( (
s  =  ( f `
 w )  /\  t  =  ( f `  z ) )  /\  w  _E  z ) }
31, 2hartogslem1 7968 . . 3  |-  ( dom 
F  C_  ~P ( A  X.  A )  /\  Fun  F  /\  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } ) )
43simp3i 1007 . 2  |-  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } )
53simp2i 1006 . . . 4  |-  Fun  F
63simp1i 1005 . . . . 5  |-  dom  F  C_ 
~P ( A  X.  A )
7 xpexg 6587 . . . . . . 7  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
87anidms 645 . . . . . 6  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
9 pwexg 4631 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
108, 9syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P ( A  X.  A
)  e.  _V )
11 ssexg 4593 . . . . 5  |-  ( ( dom  F  C_  ~P ( A  X.  A
)  /\  ~P ( A  X.  A )  e. 
_V )  ->  dom  F  e.  _V )
126, 10, 11sylancr 663 . . . 4  |-  ( A  e.  V  ->  dom  F  e.  _V )
13 funex 6129 . . . 4  |-  ( ( Fun  F  /\  dom  F  e.  _V )  ->  F  e.  _V )
145, 12, 13sylancr 663 . . 3  |-  ( A  e.  V  ->  F  e.  _V )
15 rnexg 6717 . . 3  |-  ( F  e.  _V  ->  ran  F  e.  _V )
1614, 15syl 16 . 2  |-  ( A  e.  V  ->  ran  F  e.  _V )
174, 16eqeltrrd 2556 1  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
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 5582   ` cfv 5588    ~<_ cdom 7515  OrdIsocoi 7935
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 6577
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-recs 7043  df-en 7518  df-dom 7519  df-oi 7936
This theorem is referenced by:  hartogs  7970
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