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Theorem hartogslem2 7348
Description: Lemma for hartogs 7349. (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 7347 . . 3  |-  ( dom 
F  C_  ~P ( A  X.  A )  /\  Fun  F  /\  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } ) )
43simp3i 966 . 2  |-  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } )
53simp2i 965 . . . 4  |-  Fun  F
63simp1i 964 . . . . 5  |-  dom  F  C_ 
~P ( A  X.  A )
7 xpexg 4882 . . . . . . 7  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
87anidms 626 . . . . . 6  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
9 pwexg 4275 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
108, 9syl 15 . . . . 5  |-  ( A  e.  V  ->  ~P ( A  X.  A
)  e.  _V )
11 ssexg 4241 . . . . 5  |-  ( ( dom  F  C_  ~P ( A  X.  A
)  /\  ~P ( A  X.  A )  e. 
_V )  ->  dom  F  e.  _V )
126, 10, 11sylancr 644 . . . 4  |-  ( A  e.  V  ->  dom  F  e.  _V )
13 funex 5829 . . . 4  |-  ( ( Fun  F  /\  dom  F  e.  _V )  ->  F  e.  _V )
145, 12, 13sylancr 644 . . 3  |-  ( A  e.  V  ->  F  e.  _V )
15 rnexg 5022 . . 3  |-  ( F  e.  _V  ->  ran  F  e.  _V )
1614, 15syl 15 . 2  |-  ( A  e.  V  ->  ran  F  e.  _V )
174, 16eqeltrrd 2433 1  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   E.wrex 2620   {crab 2623   _Vcvv 2864    \ cdif 3225    C_ wss 3228   ~Pcpw 3701   class class class wbr 4104   {copab 4157    _E cep 4385    _I cid 4386    We wwe 4433   Oncon0 4474    X. cxp 4769   dom cdm 4771   ran crn 4772    |` cres 4773   Fun wfun 5331   ` cfv 5337    ~<_ cdom 6949  OrdIsocoi 7314
This theorem is referenced by:  hartogs  7349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-riota 6391  df-recs 6475  df-en 6952  df-dom 6953  df-oi 7315
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