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Theorem erdszelem1 27224
Description: Lemma for erdsze 27235. (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 6226 . . . 4  |-  ( 1 ... A )  e. 
_V
21elpw2 4565 . . 3  |-  ( X  e.  ~P ( 1 ... A )  <->  X  C_  (
1 ... A ) )
32anbi1i 695 . 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 5214 . . . . . 6  |-  ( y  =  X  ->  ( F  |`  y )  =  ( F  |`  X ) )
5 isoeq1 6120 . . . . . 6  |-  ( ( F  |`  y )  =  ( F  |`  X )  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( y ,  ( F " y
) ) ) )
64, 5syl 16 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( y ,  ( F " y
) ) ) )
7 isoeq4 6123 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( y ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) ) ) )
8 imaeq2 5274 . . . . . 6  |-  ( y  =  X  ->  ( F " y )  =  ( F " X
) )
9 isoeq5 6124 . . . . . 6  |-  ( ( F " y )  =  ( F " X )  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) ) ) )
108, 9syl 16 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) ) ) )
116, 7, 103bitrd 279 . . . 4  |-  ( y  =  X  ->  (
( F  |`  y
)  Isom  <  ,  O  ( y ,  ( F " y ) )  <->  ( F  |`  X )  Isom  <  ,  O  ( X , 
( F " X
) ) ) )
12 eleq2 2527 . . . 4  |-  ( y  =  X  ->  ( A  e.  y  <->  A  e.  X ) )
1311, 12anbi12d 710 . . 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 3226 . 2  |-  ( X  e.  S  <->  ( X  e.  ~P ( 1 ... A )  /\  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) )  /\  A  e.  X
) ) )
16 3anass 969 . 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 277 1  |-  ( X  e.  S  <->  ( X  C_  ( 1 ... A
)  /\  ( F  |`  X )  Isom  <  ,  O  ( X , 
( F " X
) )  /\  A  e.  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2803    C_ wss 3437   ~Pcpw 3969    |` cres 4951   "cima 4952    Isom wiso 5528  (class class class)co 6201   1c1 9395    < clt 9530   ...cfz 11555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-ov 6204
This theorem is referenced by:  erdszelem2  27225  erdszelem4  27227  erdszelem7  27230  erdszelem8  27231
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