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Theorem cusgrasizeindslem2 23333
Description: Lemma 2 for cusgrasizeinds 23335. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
Hypothesis
Ref Expression
cusgrares.f  |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x
) } )
Assertion
Ref Expression
cusgrasizeindslem2  |-  ( dom 
F  i^i  { x  e.  dom  E  |  N  e.  ( E `  x
) } )  =  (/)
Distinct variable groups:    x, E    x, N
Allowed substitution hint:    F( x)

Proof of Theorem cusgrasizeindslem2
StepHypRef Expression
1 cusgrares.f . . . . 5  |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x
) } )
21dmeqi 5036 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
3 incom 3538 . . . . 5  |-  ( { x  e.  dom  E  |  N  e/  ( E `  x ) }  i^i  dom  E )  =  ( dom  E  i^i  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
4 dmres 5126 . . . . 5  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  ( { x  e. 
dom  E  |  N  e/  ( E `  x
) }  i^i  dom  E )
5 ssid 3370 . . . . . 6  |-  dom  E  C_ 
dom  E
6 dfrab3ss 3623 . . . . . 6  |-  ( dom 
E  C_  dom  E  ->  { x  e.  dom  E  |  N  e/  ( E `  x ) }  =  ( dom  E  i^i  { x  e. 
dom  E  |  N  e/  ( E `  x
) } ) )
75, 6ax-mp 5 . . . . 5  |-  { x  e.  dom  E  |  N  e/  ( E `  x
) }  =  ( dom  E  i^i  {
x  e.  dom  E  |  N  e/  ( E `  x ) } )
83, 4, 73eqtr4i 2468 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
92, 8eqtri 2458 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
109ineq1i 3543 . 2  |-  ( dom 
F  i^i  { x  e.  dom  E  |  N  e.  ( E `  x
) } )  =  ( { x  e. 
dom  E  |  N  e/  ( E `  x
) }  i^i  {
x  e.  dom  E  |  N  e.  ( E `  x ) } )
11 inrab 3617 . . 3  |-  ( { x  e.  dom  E  |  N  e/  ( E `  x ) }  i^i  { x  e. 
dom  E  |  N  e.  ( E `  x
) } )  =  { x  e.  dom  E  |  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x ) ) }
12 exmid 415 . . . . . . 7  |-  ( N  e.  ( E `  x )  \/  -.  N  e.  ( E `  x ) )
13 nnel 2709 . . . . . . . 8  |-  ( -.  N  e/  ( E `
 x )  <->  N  e.  ( E `  x ) )
1413orbi1i 520 . . . . . . 7  |-  ( ( -.  N  e/  ( E `  x )  \/  -.  N  e.  ( E `  x ) )  <->  ( N  e.  ( E `  x
)  \/  -.  N  e.  ( E `  x
) ) )
1512, 14mpbir 209 . . . . . 6  |-  ( -.  N  e/  ( E `
 x )  \/ 
-.  N  e.  ( E `  x ) )
16 ianor 488 . . . . . 6  |-  ( -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x )
)  <->  ( -.  N  e/  ( E `  x
)  \/  -.  N  e.  ( E `  x
) ) )
1715, 16mpbir 209 . . . . 5  |-  -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x
) )
1817rgenw 2778 . . . 4  |-  A. x  e.  dom  E  -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x
) )
19 rabeq0 3654 . . . 4  |-  ( { x  e.  dom  E  |  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x ) ) }  =  (/)  <->  A. x  e.  dom  E  -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x )
) )
2018, 19mpbir 209 . . 3  |-  { x  e.  dom  E  |  ( N  e/  ( E `
 x )  /\  N  e.  ( E `  x ) ) }  =  (/)
2111, 20eqtri 2458 . 2  |-  ( { x  e.  dom  E  |  N  e/  ( E `  x ) }  i^i  { x  e. 
dom  E  |  N  e.  ( E `  x
) } )  =  (/)
2210, 21eqtri 2458 1  |-  ( dom 
F  i^i  { x  e.  dom  E  |  N  e.  ( E `  x
) } )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    e/ wnel 2602   A.wral 2710   {crab 2714    i^i cin 3322    C_ wss 3323   (/)c0 3632   dom cdm 4835    |` cres 4837   ` cfv 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-xp 4841  df-dm 4845  df-res 4847
This theorem is referenced by:  cusgrasizeinds  23335
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