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Theorem hartogslem2 7749
Description: Lemma for hartogs 7750. (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 7748 . . 3  |-  ( dom 
F  C_  ~P ( A  X.  A )  /\  Fun  F  /\  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } ) )
43simp3i 999 . 2  |-  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } )
53simp2i 998 . . . 4  |-  Fun  F
63simp1i 997 . . . . 5  |-  dom  F  C_ 
~P ( A  X.  A )
7 xpexg 6502 . . . . . . 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 4469 . . . . . 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 4431 . . . . 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 5938 . . . 4  |-  ( ( Fun  F  /\  dom  F  e.  _V )  ->  F  e.  _V )
145, 12, 13sylancr 663 . . 3  |-  ( A  e.  V  ->  F  e.  _V )
15 rnexg 6505 . . 3  |-  ( F  e.  _V  ->  ran  F  e.  _V )
1614, 15syl 16 . 2  |-  ( A  e.  V  ->  ran  F  e.  _V )
174, 16eqeltrrd 2512 1  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2710   {crab 2713   _Vcvv 2966    \ cdif 3318    C_ wss 3321   ~Pcpw 3853   class class class wbr 4285   {copab 4342    _E cep 4622    _I cid 4623    We wwe 4670   Oncon0 4711    X. cxp 4830   dom cdm 4832   ran crn 4833    |` cres 4834   Fun wfun 5405   ` cfv 5411    ~<_ cdom 7300  OrdIsocoi 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-se 4672  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-isom 5420  df-riota 6045  df-recs 6824  df-en 7303  df-dom 7304  df-oi 7716
This theorem is referenced by:  hartogs  7750
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