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Theorem cusgrasizeindslem2 24346
Description: Lemma 2 for cusgrasizeinds 24348. (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 5194 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
3 incom 3676 . . . . 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 5284 . . . . 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 3508 . . . . . 6  |-  dom  E  C_ 
dom  E
6 dfrab3ss 3761 . . . . . 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 2482 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
92, 8eqtri 2472 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
109ineq1i 3681 . 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 3755 . . 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 2788 . . . . . . . 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 2804 . . . 4  |-  A. x  e.  dom  E  -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x
) )
19 rabeq0 3793 . . . 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 2472 . 2  |-  ( { x  e.  dom  E  |  N  e/  ( E `  x ) }  i^i  { x  e. 
dom  E  |  N  e.  ( E `  x
) } )  =  (/)
2210, 21eqtri 2472 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 1383    e. wcel 1804    e/ wnel 2639   A.wral 2793   {crab 2797    i^i cin 3460    C_ wss 3461   (/)c0 3770   dom cdm 4989    |` cres 4991   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-dm 4999  df-res 5001
This theorem is referenced by:  cusgrasizeinds  24348
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