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Theorem hartogslem2 7476
Description: Lemma for hartogs 7477. (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 7475 . . 3  |-  ( dom 
F  C_  ~P ( A  X.  A )  /\  Fun  F  /\  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } ) )
43simp3i 968 . 2  |-  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } )
53simp2i 967 . . . 4  |-  Fun  F
63simp1i 966 . . . . 5  |-  dom  F  C_ 
~P ( A  X.  A )
7 xpexg 4956 . . . . . . 7  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
87anidms 627 . . . . . 6  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
9 pwexg 4351 . . . . . 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 4317 . . . . 5  |-  ( ( dom  F  C_  ~P ( A  X.  A
)  /\  ~P ( A  X.  A )  e. 
_V )  ->  dom  F  e.  _V )
126, 10, 11sylancr 645 . . . 4  |-  ( A  e.  V  ->  dom  F  e.  _V )
13 funex 5930 . . . 4  |-  ( ( Fun  F  /\  dom  F  e.  _V )  ->  F  e.  _V )
145, 12, 13sylancr 645 . . 3  |-  ( A  e.  V  ->  F  e.  _V )
15 rnexg 5098 . . 3  |-  ( F  e.  _V  ->  ran  F  e.  _V )
1614, 15syl 16 . 2  |-  ( A  e.  V  ->  ran  F  e.  _V )
174, 16eqeltrrd 2487 1  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675   {crab 2678   _Vcvv 2924    \ cdif 3285    C_ wss 3288   ~Pcpw 3767   class class class wbr 4180   {copab 4233    _E cep 4460    _I cid 4461    We wwe 4508   Oncon0 4549    X. cxp 4843   dom cdm 4845   ran crn 4846    |` cres 4847   Fun wfun 5415   ` cfv 5421    ~<_ cdom 7074  OrdIsocoi 7442
This theorem is referenced by:  hartogs  7477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6516  df-recs 6600  df-en 7077  df-dom 7078  df-oi 7443
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