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Theorem erdszelem1 28832
Description: Lemma for erdsze 28843. (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 6324 . . . 4  |-  ( 1 ... A )  e. 
_V
21elpw2 4620 . . 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 5278 . . . . . 6  |-  ( y  =  X  ->  ( F  |`  y )  =  ( F  |`  X ) )
5 isoeq1 6216 . . . . . 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 6219 . . . . 5  |-  ( y  =  X  ->  (
( F  |`  X ) 
Isom  <  ,  O  ( y ,  ( F
" y ) )  <-> 
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" y ) ) ) )
8 imaeq2 5343 . . . . . 6  |-  ( y  =  X  ->  ( F " y )  =  ( F " X
) )
9 isoeq5 6220 . . . . . 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 2530 . . . 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 3259 . 2  |-  ( X  e.  S  <->  ( X  e.  ~P ( 1 ... A )  /\  (
( F  |`  X ) 
Isom  <  ,  O  ( X ,  ( F
" X ) )  /\  A  e.  X
) ) )
16 3anass 977 . 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 973    = wceq 1395    e. wcel 1819   {crab 2811    C_ wss 3471   ~Pcpw 4015    |` cres 5010   "cima 5011    Isom wiso 5595  (class class class)co 6296   1c1 9510    < clt 9645   ...cfz 11697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-ov 6299
This theorem is referenced by:  erdszelem2  28833  erdszelem4  28835  erdszelem7  28838  erdszelem8  28839
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