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Theorem hartogslem1 7257
Description: Lemma for hartogs 7259. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
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
hartogslem1  |-  ( dom 
F  C_  ~P ( A  X.  A )  /\  Fun  F  /\  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } ) )
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 hartogslem1
StepHypRef Expression
1 hartogslem.2 . . . . 5  |-  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 ) ) }
21dmeqi 4880 . . . 4  |-  dom  F  =  dom  { <. 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 ) ) }
3 dmopab 4889 . . . 4  |-  dom  { <. 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 ) ) }  =  { r  |  E. 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 ) ) }
42, 3eqtri 2303 . . 3  |-  dom  F  =  { r  |  E. 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 ) ) }
5 simp3 957 . . . . . . . 8  |-  ( ( dom  r  C_  A  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  ->  r  C_  ( dom  r  X.  dom  r
) )
6 simp1 955 . . . . . . . . 9  |-  ( ( dom  r  C_  A  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  ->  dom  r  C_  A )
7 xpss12 4792 . . . . . . . . 9  |-  ( ( dom  r  C_  A  /\  dom  r  C_  A
)  ->  ( dom  r  X.  dom  r ) 
C_  ( A  X.  A ) )
86, 6, 7syl2anc 642 . . . . . . . 8  |-  ( ( dom  r  C_  A  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  ->  ( dom  r  X.  dom  r )  C_  ( A  X.  A
) )
95, 8sstrd 3189 . . . . . . 7  |-  ( ( dom  r  C_  A  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  ->  r  C_  ( A  X.  A ) )
10 vex 2791 . . . . . . . 8  |-  r  e. 
_V
1110elpw 3631 . . . . . . 7  |-  ( r  e.  ~P ( A  X.  A )  <->  r  C_  ( A  X.  A
) )
129, 11sylibr 203 . . . . . 6  |-  ( ( dom  r  C_  A  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  ->  r  e.  ~P ( A  X.  A
) )
1312ad2antrr 706 . . . . 5  |-  ( ( ( ( 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 ) )  ->  r  e.  ~P ( A  X.  A
) )
1413exlimiv 1666 . . . 4  |-  ( E. 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 ) )  ->  r  e.  ~P ( A  X.  A
) )
1514abssi 3248 . . 3  |-  { r  |  E. 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 ) ) }  C_  ~P ( A  X.  A )
164, 15eqsstri 3208 . 2  |-  dom  F  C_ 
~P ( A  X.  A )
17 funopab4 5289 . . 3  |-  Fun  { <. 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 ) ) }
181funeqi 5275 . . 3  |-  ( Fun 
F  <->  Fun  { <. 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 ) ) } )
1917, 18mpbir 200 . 2  |-  Fun  F
20 breq1 4026 . . . . . 6  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
2120elrab 2923 . . . . 5  |-  ( y  e.  { x  e.  On  |  x  ~<_  A }  <->  ( y  e.  On  /\  y  ~<_  A ) )
22 brdomi 6873 . . . . . . 7  |-  ( y  ~<_  A  ->  E. f 
f : y -1-1-> A
)
23 f1f 5437 . . . . . . . . . . . . . 14  |-  ( f : y -1-1-> A  -> 
f : y --> A )
2423adantl 452 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  f : y --> A )
25 frn 5395 . . . . . . . . . . . . 13  |-  ( f : y --> A  ->  ran  f  C_  A )
2624, 25syl 15 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ran  f  C_  A )
27 resss 4979 . . . . . . . . . . . . . . 15  |-  (  _I  |`  ran  f )  C_  _I
28 ssun2 3339 . . . . . . . . . . . . . . 15  |-  _I  C_  ( R  u.  _I  )
2927, 28sstri 3188 . . . . . . . . . . . . . 14  |-  (  _I  |`  ran  f )  C_  ( R  u.  _I  )
30 f1oi 5511 . . . . . . . . . . . . . . 15  |-  (  _I  |`  ran  f ) : ran  f -1-1-onto-> ran  f
31 f1of 5472 . . . . . . . . . . . . . . 15  |-  ( (  _I  |`  ran  f ) : ran  f -1-1-onto-> ran  f  ->  (  _I  |`  ran  f
) : ran  f --> ran  f )
32 fssxp 5400 . . . . . . . . . . . . . . 15  |-  ( (  _I  |`  ran  f ) : ran  f --> ran  f  ->  (  _I  |` 
ran  f )  C_  ( ran  f  X.  ran  f ) )
3330, 31, 32mp2b 9 . . . . . . . . . . . . . 14  |-  (  _I  |`  ran  f )  C_  ( ran  f  X.  ran  f )
3429, 33ssini 3392 . . . . . . . . . . . . 13  |-  (  _I  |`  ran  f )  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X.  ran  f
) )
3534a1i 10 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  (  _I  |`  ran  f
)  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) )
36 inss2 3390 . . . . . . . . . . . . 13  |-  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) 
C_  ( ran  f  X.  ran  f )
3736a1i 10 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) 
C_  ( ran  f  X.  ran  f ) )
3826, 35, 373jca 1132 . . . . . . . . . . 11  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( ran  f  C_  A  /\  (  _I  |`  ran  f )  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X.  ran  f
) )  /\  (
( R  u.  _I  )  i^i  ( ran  f  X.  ran  f ) ) 
C_  ( ran  f  X.  ran  f ) ) )
39 eloni 4402 . . . . . . . . . . . . . . 15  |-  ( y  e.  On  ->  Ord  y )
40 ordwe 4405 . . . . . . . . . . . . . . 15  |-  ( Ord  y  ->  _E  We  y )
4139, 40syl 15 . . . . . . . . . . . . . 14  |-  ( y  e.  On  ->  _E  We  y )
4241adantr 451 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  _E  We  y
)
43 f1f1orn 5483 . . . . . . . . . . . . . . . . 17  |-  ( f : y -1-1-> A  -> 
f : y -1-1-onto-> ran  f
)
4443adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  f : y -1-1-onto-> ran  f )
45 hartogslem.3 . . . . . . . . . . . . . . . 16  |-  R  =  { <. s ,  t
>.  |  E. w  e.  y  E. z  e.  y  ( (
s  =  ( f `
 w )  /\  t  =  ( f `  z ) )  /\  w  _E  z ) }
46 f1oiso 5848 . . . . . . . . . . . . . . . 16  |-  ( ( f : y -1-1-onto-> ran  f  /\  R  =  { <. s ,  t >.  |  E. w  e.  y  E. z  e.  y  ( ( s  =  ( f `  w
)  /\  t  =  ( f `  z
) )  /\  w  _E  z ) } )  ->  f  Isom  _E  ,  R  ( y ,  ran  f ) )
4744, 45, 46sylancl 643 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  f  Isom  _E  ,  R  ( y ,  ran  f ) )
48 isores2 5830 . . . . . . . . . . . . . . 15  |-  ( f 
Isom  _E  ,  R  ( y ,  ran  f )  <->  f  Isom  _E  ,  ( R  i^i  ( ran  f  X.  ran  f ) ) ( y ,  ran  f
) )
4947, 48sylib 188 . . . . . . . . . . . . . 14  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  f  Isom  _E  , 
( R  i^i  ( ran  f  X.  ran  f
) ) ( y ,  ran  f ) )
50 isowe 5846 . . . . . . . . . . . . . 14  |-  ( f 
Isom  _E  ,  ( R  i^i  ( ran  f  X.  ran  f ) ) ( y ,  ran  f )  ->  (  _E  We  y  <->  ( R  i^i  ( ran  f  X. 
ran  f ) )  We  ran  f ) )
5149, 50syl 15 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  (  _E  We  y 
<->  ( R  i^i  ( ran  f  X.  ran  f
) )  We  ran  f ) )
5242, 51mpbid 201 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( R  i^i  ( ran  f  X.  ran  f ) )  We 
ran  f )
53 weso 4384 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( R  i^i  ( ran  f  X.  ran  f
) )  We  ran  f  ->  ( R  i^i  ( ran  f  X.  ran  f ) )  Or 
ran  f )
5452, 53syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( R  i^i  ( ran  f  X.  ran  f ) )  Or 
ran  f )
55 inss2 3390 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  i^i  ( ran  f  X.  ran  f ) ) 
C_  ( ran  f  X.  ran  f )
5655brel 4737 . . . . . . . . . . . . . . . . . . . 20  |-  ( x ( R  i^i  ( ran  f  X.  ran  f
) ) x  -> 
( x  e.  ran  f  /\  x  e.  ran  f ) )
5756simpld 445 . . . . . . . . . . . . . . . . . . 19  |-  ( x ( R  i^i  ( ran  f  X.  ran  f
) ) x  ->  x  e.  ran  f )
58 sonr 4335 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  Or  ran  f  /\  x  e.  ran  f )  ->  -.  x ( R  i^i  ( ran  f  X.  ran  f ) ) x )
5954, 57, 58syl2an 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  On  /\  f : y -1-1-> A
)  /\  x ( R  i^i  ( ran  f  X.  ran  f ) ) x )  ->  -.  x ( R  i^i  ( ran  f  X.  ran  f ) ) x )
6059ex 423 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( x ( R  i^i  ( ran  f  X.  ran  f
) ) x  ->  -.  x ( R  i^i  ( ran  f  X.  ran  f ) ) x ) )
6160pm2.01d 161 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  -.  x ( R  i^i  ( ran  f  X.  ran  f ) ) x )
6261alrimiv 1617 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  A. x  -.  x
( R  i^i  ( ran  f  X.  ran  f
) ) x )
63 intirr 5061 . . . . . . . . . . . . . . 15  |-  ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  i^i  _I  )  =  (/)  <->  A. x  -.  x ( R  i^i  ( ran  f  X.  ran  f ) ) x )
6462, 63sylibr 203 . . . . . . . . . . . . . 14  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( ( R  i^i  ( ran  f  X.  ran  f ) )  i^i  _I  )  =  (/) )
65 disj3 3499 . . . . . . . . . . . . . 14  |-  ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  i^i  _I  )  =  (/)  <->  ( R  i^i  ( ran  f  X. 
ran  f ) )  =  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  ) )
6664, 65sylib 188 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( R  i^i  ( ran  f  X.  ran  f ) )  =  ( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) )
67 weeq1 4381 . . . . . . . . . . . . 13  |-  ( ( R  i^i  ( ran  f  X.  ran  f
) )  =  ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  )  ->  ( ( R  i^i  ( ran  f  X.  ran  f ) )  We  ran  f  <->  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f ) )
6866, 67syl 15 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( ( R  i^i  ( ran  f  X.  ran  f ) )  We  ran  f  <->  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f ) )
6952, 68mpbid 201 . . . . . . . . . . 11  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f )
7039adantr 451 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  Ord  y )
71 isoeq3 5818 . . . . . . . . . . . . . . 15  |-  ( ( R  i^i  ( ran  f  X.  ran  f
) )  =  ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  )  ->  ( f  Isom  _E  ,  ( R  i^i  ( ran  f  X.  ran  f ) ) ( y ,  ran  f
)  <->  f  Isom  _E  , 
(  ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) ( y ,  ran  f ) ) )
7266, 71syl 15 . . . . . . . . . . . . . 14  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( f  Isom  _E  ,  ( R  i^i  ( ran  f  X.  ran  f ) ) ( y ,  ran  f
)  <->  f  Isom  _E  , 
(  ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) ( y ,  ran  f ) ) )
7349, 72mpbid 201 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  f  Isom  _E  , 
(  ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) ( y ,  ran  f ) )
74 vex 2791 . . . . . . . . . . . . . . . 16  |-  f  e. 
_V
7574rnex 4942 . . . . . . . . . . . . . . 15  |-  ran  f  e.  _V
76 exse 4357 . . . . . . . . . . . . . . 15  |-  ( ran  f  e.  _V  ->  ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) Se  ran  f )
7775, 76ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) Se  ran  f
78 eqid 2283 . . . . . . . . . . . . . . 15  |- OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  ran  f )  = OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  ran  f )
7978oieu 7254 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  )  We  ran  f  /\  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  ) Se  ran  f )  ->  (
( Ord  y  /\  f  Isom  _E  ,  (  ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ( y ,  ran  f ) )  <-> 
( y  =  dom OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) ,  ran  f
)  /\  f  = OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) ,  ran  f
) ) ) )
8069, 77, 79sylancl 643 . . . . . . . . . . . . 13  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( ( Ord  y  /\  f  Isom  _E  ,  (  ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  ) ( y ,  ran  f
) )  <->  ( y  =  dom OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  ran  f )  /\  f  = OrdIso (
( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) ,  ran  f
) ) ) )
8170, 73, 80mpbi2and 887 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  ( y  =  dom OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  ran  f )  /\  f  = OrdIso (
( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) ,  ran  f
) ) )
8281simpld 445 . . . . . . . . . . 11  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  y  =  dom OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  ) ,  ran  f
) )
8375, 75xpex 4801 . . . . . . . . . . . . 13  |-  ( ran  f  X.  ran  f
)  e.  _V
8483inex2 4156 . . . . . . . . . . . 12  |-  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  e.  _V
85 sseq1 3199 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( r  C_  ( ran  f  X.  ran  f )  <->  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) 
C_  ( ran  f  X.  ran  f ) ) )
8636, 85mpbiri 224 . . . . . . . . . . . . . . . . . . 19  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  r  C_  ( ran  f  X.  ran  f
) )
87 dmss 4878 . . . . . . . . . . . . . . . . . . 19  |-  ( r 
C_  ( ran  f  X.  ran  f )  ->  dom  r  C_  dom  ( ran  f  X.  ran  f
) )
8886, 87syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  dom  r  C_  dom  ( ran  f  X. 
ran  f ) )
89 dmxpid 4898 . . . . . . . . . . . . . . . . . 18  |-  dom  ( ran  f  X.  ran  f
)  =  ran  f
9088, 89syl6sseq 3224 . . . . . . . . . . . . . . . . 17  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  dom  r  C_  ran  f )
91 dmresi 5005 . . . . . . . . . . . . . . . . . 18  |-  dom  (  _I  |`  ran  f )  =  ran  f
92 sseq2 3200 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( (  _I  |`  ran  f )  C_  r 
<->  (  _I  |`  ran  f
)  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) ) )
9334, 92mpbiri 224 . . . . . . . . . . . . . . . . . . 19  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  (  _I  |`  ran  f
)  C_  r )
94 dmss 4878 . . . . . . . . . . . . . . . . . . 19  |-  ( (  _I  |`  ran  f ) 
C_  r  ->  dom  (  _I  |`  ran  f
)  C_  dom  r )
9593, 94syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  dom  (  _I  |` 
ran  f )  C_  dom  r )
9691, 95syl5eqssr 3223 . . . . . . . . . . . . . . . . 17  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ran  f  C_  dom  r )
9790, 96eqssd 3196 . . . . . . . . . . . . . . . 16  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  dom  r  =  ran  f )
9897sseq1d 3205 . . . . . . . . . . . . . . 15  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( dom  r  C_  A  <->  ran  f  C_  A
) )
9997reseq2d 4955 . . . . . . . . . . . . . . . 16  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  (  _I  |`  dom  r
)  =  (  _I  |`  ran  f ) )
100 id 19 . . . . . . . . . . . . . . . 16  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X.  ran  f ) ) )
10199, 100sseq12d 3207 . . . . . . . . . . . . . . 15  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( (  _I  |`  dom  r )  C_  r 
<->  (  _I  |`  ran  f
)  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) ) )
10297, 97xpeq12d 4714 . . . . . . . . . . . . . . . 16  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( dom  r  X.  dom  r )  =  ( ran  f  X. 
ran  f ) )
103100, 102sseq12d 3207 . . . . . . . . . . . . . . 15  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( r  C_  ( dom  r  X.  dom  r )  <->  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) 
C_  ( ran  f  X.  ran  f ) ) )
10498, 101, 1033anbi123d 1252 . . . . . . . . . . . . . 14  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( ( dom  r  C_  A  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  <-> 
( ran  f  C_  A  /\  (  _I  |`  ran  f
)  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  /\  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) 
C_  ( ran  f  X.  ran  f ) ) ) )
105 difeq1 3287 . . . . . . . . . . . . . . . . 17  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( r  \  _I  )  =  (
( ( R  u.  _I  )  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) )
106 difun2 3533 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  u.  _I  )  \  _I  )  =  ( R  \  _I  )
107106ineq1i 3366 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  u.  _I  )  \  _I  )  i^i  ( ran  f  X. 
ran  f ) )  =  ( ( R 
\  _I  )  i^i  ( ran  f  X. 
ran  f ) )
108 indif1 3413 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  u.  _I  )  \  _I  )  i^i  ( ran  f  X. 
ran  f ) )  =  ( ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) 
\  _I  )
109 indif1 3413 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  \  _I  )  i^i  ( ran  f  X. 
ran  f ) )  =  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )
110107, 108, 1093eqtr3i 2311 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  u.  _I  )  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  =  ( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  )
111105, 110syl6eq 2331 . . . . . . . . . . . . . . . 16  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( r  \  _I  )  =  (
( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) )
112 weeq1 4381 . . . . . . . . . . . . . . . 16  |-  ( ( r  \  _I  )  =  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  -> 
( ( r  \  _I  )  We  dom  r 
<->  ( ( R  i^i  ( ran  f  X.  ran  f ) )  \  _I  )  We  dom  r ) )
113111, 112syl 15 . . . . . . . . . . . . . . 15  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( ( r 
\  _I  )  We 
dom  r  <->  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
dom  r ) )
114 weeq2 4382 . . . . . . . . . . . . . . . 16  |-  ( dom  r  =  ran  f  ->  ( ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
dom  r  <->  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f ) )
11597, 114syl 15 . . . . . . . . . . . . . . 15  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  )  We  dom  r  <->  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f ) )
116113, 115bitrd 244 . . . . . . . . . . . . . 14  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( ( r 
\  _I  )  We 
dom  r  <->  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f ) )
117104, 116anbi12d 691 . . . . . . . . . . . . 13  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( ( ( dom  r  C_  A  /\  (  _I  |`  dom  r
)  C_  r  /\  r  C_  ( dom  r  X.  dom  r ) )  /\  ( r  \  _I  )  We  dom  r )  <->  ( ( ran  f  C_  A  /\  (  _I  |`  ran  f
)  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  /\  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) 
C_  ( ran  f  X.  ran  f ) )  /\  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f ) ) )
118 oieq1 7227 . . . . . . . . . . . . . . . . 17  |-  ( ( r  \  _I  )  =  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  -> OrdIso ( ( r  \  _I  ) ,  dom  r
)  = OrdIso ( (
( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  dom  r ) )
119111, 118syl 15 . . . . . . . . . . . . . . . 16  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  -> OrdIso ( ( r  \  _I  ) ,  dom  r
)  = OrdIso ( (
( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  dom  r ) )
120 oieq2 7228 . . . . . . . . . . . . . . . . 17  |-  ( dom  r  =  ran  f  -> OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  ) ,  dom  r )  = OrdIso
( ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  ) ,  ran  f ) )
12197, 120syl 15 . . . . . . . . . . . . . . . 16  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  -> OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  ) ,  dom  r )  = OrdIso
( ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  ) ,  ran  f ) )
122119, 121eqtrd 2315 . . . . . . . . . . . . . . 15  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  -> OrdIso ( ( r  \  _I  ) ,  dom  r
)  = OrdIso ( (
( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  ran  f ) )
123122dmeqd 4881 . . . . . . . . . . . . . 14  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  dom OrdIso ( (
r  \  _I  ) ,  dom  r )  =  dom OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  ran  f ) )
124123eqeq2d 2294 . . . . . . . . . . . . 13  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( y  =  dom OrdIso ( ( r 
\  _I  ) ,  dom  r )  <->  y  =  dom OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  ) ,  ran  f ) ) )
125117, 124anbi12d 691 . . . . . . . . . . . 12  |-  ( r  =  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  ->  ( ( ( ( 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 ) )  <-> 
( ( ( ran  f  C_  A  /\  (  _I  |`  ran  f
)  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) )  /\  ( ( R  u.  _I  )  i^i  ( ran  f  X. 
ran  f ) ) 
C_  ( ran  f  X.  ran  f ) )  /\  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f )  /\  y  =  dom OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  ran  f ) ) ) )
12684, 125spcev 2875 . . . . . . . . . . 11  |-  ( ( ( ( ran  f  C_  A  /\  (  _I  |`  ran  f )  C_  ( ( R  u.  _I  )  i^i  ( ran  f  X.  ran  f
) )  /\  (
( R  u.  _I  )  i^i  ( ran  f  X.  ran  f ) ) 
C_  ( ran  f  X.  ran  f ) )  /\  ( ( R  i^i  ( ran  f  X.  ran  f ) ) 
\  _I  )  We 
ran  f )  /\  y  =  dom OrdIso ( ( ( R  i^i  ( ran  f  X.  ran  f
) )  \  _I  ) ,  ran  f ) )  ->  E. r
( ( ( 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 ) ) )
12738, 69, 82, 126syl21anc 1181 . . . . . . . . . 10  |-  ( ( y  e.  On  /\  f : y -1-1-> A )  ->  E. r ( ( ( 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 ) ) )
128127ex 423 . . . . . . . . 9  |-  ( y  e.  On  ->  (
f : y -1-1-> A  ->  E. r ( ( ( 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 ) ) ) )
129128exlimdv 1664 . . . . . . . 8  |-  ( y  e.  On  ->  ( E. f  f :
y -1-1-> A  ->  E. r
( ( ( 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 ) ) ) )
130129imp 418 . . . . . . 7  |-  ( ( y  e.  On  /\  E. f  f : y
-1-1-> A )  ->  E. r
( ( ( 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 ) ) )
13122, 130sylan2 460 . . . . . 6  |-  ( ( y  e.  On  /\  y  ~<_  A )  ->  E. r ( ( ( 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 ) ) )
132 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( 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 ) )  ->  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) )
13310dmex 4941 . . . . . . . . . . . 12  |-  dom  r  e.  _V
134 eqid 2283 . . . . . . . . . . . . 13  |- OrdIso ( ( r  \  _I  ) ,  dom  r )  = OrdIso
( ( r  \  _I  ) ,  dom  r
)
135134oion 7251 . . . . . . . . . . . 12  |-  ( dom  r  e.  _V  ->  dom OrdIso ( ( r  \  _I  ) ,  dom  r
)  e.  On )
136133, 135ax-mp 8 . . . . . . . . . . 11  |-  dom OrdIso ( ( r  \  _I  ) ,  dom  r )  e.  On
137132, 136syl6eqel 2371 . . . . . . . . . 10  |-  ( ( ( ( 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 ) )  ->  y  e.  On )
138137adantl 452 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( ( ( 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 ) ) )  ->  y  e.  On )
139 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) )  ->  ( r  \  _I  )  We  dom  r )
140134oien 7253 . . . . . . . . . . . . 13  |-  ( ( dom  r  e.  _V  /\  ( r  \  _I  )  We  dom  r )  ->  dom OrdIso ( (
r  \  _I  ) ,  dom  r )  ~~  dom  r )
141133, 139, 140sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) )  ->  dom OrdIso ( (
r  \  _I  ) ,  dom  r )  ~~  dom  r )
142132, 141eqbrtrd 4043 . . . . . . . . . . 11  |-  ( ( ( ( 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 ) )  ->  y  ~~  dom  r )
143142adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( ( ( 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 ) ) )  ->  y  ~~  dom  r )
144 simpll1 994 . . . . . . . . . . 11  |-  ( ( ( ( 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 ) )  ->  dom  r  C_  A )
145 ssdomg 6907 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( dom  r  C_  A  ->  dom  r  ~<_  A )
)
146145imp 418 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  dom  r  C_  A )  ->  dom  r  ~<_  A )
147144, 146sylan2 460 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( ( ( 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 ) ) )  ->  dom  r  ~<_  A )
148 endomtr 6919 . . . . . . . . . 10  |-  ( ( y  ~~  dom  r  /\  dom  r  ~<_  A )  ->  y  ~<_  A )
149143, 147, 148syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( ( ( 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 ) ) )  ->  y  ~<_  A )
150138, 149jca 518 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( ( ( 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 ) ) )  ->  ( y  e.  On  /\  y  ~<_  A ) )
151150ex 423 . . . . . . 7  |-  ( A  e.  V  ->  (
( ( ( 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 ) )  ->  ( y  e.  On  /\  y  ~<_  A ) ) )
152151exlimdv 1664 . . . . . 6  |-  ( A  e.  V  ->  ( E. r ( ( ( 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 ) )  ->  ( y  e.  On  /\  y  ~<_  A ) ) )
153131, 152impbid2 195 . . . . 5  |-  ( A  e.  V  ->  (
( y  e.  On  /\  y  ~<_  A )  <->  E. r
( ( ( 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 ) ) ) )
15421, 153syl5bb 248 . . . 4  |-  ( A  e.  V  ->  (
y  e.  { x  e.  On  |  x  ~<_  A }  <->  E. r ( ( ( 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 ) ) ) )
155154abbi2dv 2398 . . 3  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  =  { y  |  E. r ( ( ( 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 ) ) } )
1561rneqi 4905 . . . 4  |-  ran  F  =  ran  { <. 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 ) ) }
157 rnopab 4924 . . . 4  |-  ran  { <. 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 ) ) }  =  { y  |  E. r ( ( ( 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 ) ) }
158156, 157eqtri 2303 . . 3  |-  ran  F  =  { y  |  E. r ( ( ( 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 ) ) }
159155, 158syl6reqr 2334 . 2  |-  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } )
16016, 19, 1593pm3.2i 1130 1  |-  ( dom 
F  C_  ~P ( A  X.  A )  /\  Fun  F  /\  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023   {copab 4076    _E cep 4303    _I cid 4304    Or wor 4313   Se wse 4350    We wwe 4351   Ord word 4391   Oncon0 4392    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256    ~~ cen 6860    ~<_ cdom 6861  OrdIsocoi 7224
This theorem is referenced by:  hartogslem2  7258  harwdom  7304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-en 6864  df-dom 6865  df-oi 7225
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