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Theorem erdszelem1 23722
Description: Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypothesis
Ref Expression
erdszelem1.1  |-  S  =  { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) }
Assertion
Ref Expression
erdszelem1  |-  ( X  e.  S  <->  ( X  C_  ( 1 ... A
)  /\  ( F  |`  X )  Isom  <  ,  O  ( X , 
( F " X
) )  /\  A  e.  X ) )
Distinct variable groups:    y, A    y, F    y, O    y, X
Allowed substitution hint:    S( y)

Proof of Theorem erdszelem1
StepHypRef Expression
1 ovex 5883 . . . 4  |-  ( 1 ... A )  e. 
_V
21elpw2 4175 . . 3  |-  ( X  e.  ~P ( 1 ... A )  <->  X  C_  (
1 ... A ) )
32anbi1i 676 . 2  |-  ( ( X  e.  ~P (
1 ... A )  /\  ( ( F  |`  X )  Isom  <  ,  O  ( X , 
( F " X
) )  /\  A  e.  X ) )  <->  ( X  C_  ( 1 ... A
)  /\  ( ( F  |`  X )  Isom  <  ,  O  ( X ,  ( F " X ) )  /\  A  e.  X )
) )
4 reseq2 4950 . . . . . 6  |-  ( y  =  X  ->  ( F  |`  y )  =  ( F  |`  X ) )
5 isoeq1 5816 . . . . . 6  |-  ( ( F  |`  y )  =  ( F  |`  X )  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( y ,  ( F " y
) ) ) )
64, 5syl 15 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( y ,  ( F " y
) ) ) )
7 isoeq4 5819 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( y ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) ) ) )
8 imaeq2 5008 . . . . . 6  |-  ( y  =  X  ->  ( F " y )  =  ( F " X
) )
9 isoeq5 5820 . . . . . 6  |-  ( ( F " y )  =  ( F " X )  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) ) ) )
108, 9syl 15 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) ) ) )
116, 7, 103bitrd 270 . . . 4  |-  ( y  =  X  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( X , 
( F " X
) ) ) )
12 eleq2 2344 . . . 4  |-  ( y  =  X  ->  ( A  e.  y  <->  A  e.  X ) )
1311, 12anbi12d 691 . . 3  |-  ( y  =  X  ->  (
( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
) )  /\  A  e.  y )  <->  ( ( F  |`  X )  Isom  <  ,  O  ( X ,  ( F " X ) )  /\  A  e.  X )
) )
14 erdszelem1.1 . . 3  |-  S  =  { y  e.  ~P ( 1 ... A
)  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F
" y ) )  /\  A  e.  y ) }
1513, 14elrab2 2925 . 2  |-  ( X  e.  S  <->  ( X  e.  ~P ( 1 ... A )  /\  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) )  /\  A  e.  X
) ) )
16 3anass 938 . 2  |-  ( ( X  C_  ( 1 ... A )  /\  ( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) )  /\  A  e.  X
)  <->  ( X  C_  ( 1 ... A
)  /\  ( ( F  |`  X )  Isom  <  ,  O  ( X ,  ( F " X ) )  /\  A  e.  X )
) )
173, 15, 163bitr4i 268 1  |-  ( X  e.  S  <->  ( X  C_  ( 1 ... A
)  /\  ( F  |`  X )  Isom  <  ,  O  ( X , 
( F " X
) )  /\  A  e.  X ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ~Pcpw 3625    |` cres 4691   "cima 4692    Isom wiso 5256  (class class class)co 5858   1c1 8738    < clt 8867   ...cfz 10782
This theorem is referenced by:  erdszelem2  23723  erdszelem4  23725  erdszelem7  23728  erdszelem8  23729
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  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-ov 5861
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