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Theorem cusgrasizeindslem2 23519
Description: Lemma 2 for cusgrasizeinds 23521. (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 5141 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
3 incom 3643 . . . . 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 5231 . . . . 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 3475 . . . . . 6  |-  dom  E  C_ 
dom  E
6 dfrab3ss 3728 . . . . . 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 2490 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
92, 8eqtri 2480 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
109ineq1i 3648 . 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 3722 . . 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 2793 . . . . . . . 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 2893 . . . 4  |-  A. x  e.  dom  E  -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x
) )
19 rabeq0 3759 . . . 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 2480 . 2  |-  ( { x  e.  dom  E  |  N  e/  ( E `  x ) }  i^i  { x  e. 
dom  E  |  N  e.  ( E `  x
) } )  =  (/)
2210, 21eqtri 2480 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 1370    e. wcel 1758    e/ wnel 2645   A.wral 2795   {crab 2799    i^i cin 3427    C_ wss 3428   (/)c0 3737   dom cdm 4940    |` cres 4942   ` cfv 5518
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
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-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-xp 4946  df-dm 4950  df-res 4952
This theorem is referenced by:  cusgrasizeinds  23521
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