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Theorem erdszelem1 22893
Description: Lemma for erdsze 22904. (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 5735 . . . 4  |-  ( 1 ... A )  e. 
_V
21elpw2 4064 . . 3  |-  ( X  e.  ~P ( 1 ... A )  <->  X  C_  (
1 ... A ) )
32anbi1i 679 . 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 4857 . . . . . 6  |-  ( y  =  X  ->  ( F  |`  y )  =  ( F  |`  X ) )
5 isoeq1 5668 . . . . . 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 5671 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( y ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) ) ) )
8 imaeq2 4915 . . . . . 6  |-  ( y  =  X  ->  ( F " y )  =  ( F " X
) )
9 isoeq5 5672 . . . . . 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 2314 . . . 4  |-  ( y  =  X  ->  ( A  e.  y  <->  A  e.  X ) )
1311, 12anbi12d 694 . . 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 2862 . 2  |-  ( X  e.  S  <->  ( X  e.  ~P ( 1 ... A )  /\  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) )  /\  A  e.  X
) ) )
16 3anass 943 . 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 939    = wceq 1619    e. wcel 1621   {crab 2512    C_ wss 3078   ~Pcpw 3530    |` cres 4582   "cima 4583    Isom wiso 4593  (class class class)co 5710   1c1 8618    < clt 8747   ...cfz 10660
This theorem is referenced by:  erdszelem2  22894  erdszelem4  22896  erdszelem7  22899  erdszelem8  22900
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713
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