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Theorem erdszelem1 23126
Description: Lemma for erdsze 23137. (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 5844 . . . 4  |-  ( 1 ... A )  e. 
_V
21elpw2 4169 . . 3  |-  ( X  e.  ~P ( 1 ... A )  <->  X  C_  (
1 ... A ) )
32anbi1i 678 . 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 4949 . . . . . 6  |-  ( y  =  X  ->  ( F  |`  y )  =  ( F  |`  X ) )
5 isoeq1 5777 . . . . . 6  |-  ( ( F  |`  y )  =  ( F  |`  X )  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( y ,  ( F " y
) ) ) )
64, 5syl 17 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( y ,  ( F " y
) ) ) )
7 isoeq4 5780 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( y ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) ) ) )
8 imaeq2 5007 . . . . . 6  |-  ( y  =  X  ->  ( F " y )  =  ( F " X
) )
9 isoeq5 5781 . . . . . 6  |-  ( ( F " y )  =  ( F " X )  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) ) ) )
108, 9syl 17 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) ) ) )
116, 7, 103bitrd 272 . . . 4  |-  ( y  =  X  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( X , 
( F " X
) ) ) )
12 eleq2 2345 . . . 4  |-  ( y  =  X  ->  ( A  e.  y  <->  A  e.  X ) )
1311, 12anbi12d 693 . . 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 2926 . 2  |-  ( X  e.  S  <->  ( X  e.  ~P ( 1 ... A )  /\  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) )  /\  A  e.  X
) ) )
16 3anass 940 . 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 270 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 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   {crab 2548    C_ wss 3153   ~Pcpw 3626    |` cres 4690   "cima 4691    Isom wiso 5222  (class class class)co 5819   1c1 8733    < clt 8862   ...cfz 10776
This theorem is referenced by:  erdszelem2  23127  erdszelem4  23129  erdszelem7  23132  erdszelem8  23133
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822
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