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Theorem cusgrasizeindslem2 24150
Description: Lemma 2 for cusgrasizeinds 24152. (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 5202 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
3 incom 3691 . . . . 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 5292 . . . . 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 3523 . . . . . 6  |-  dom  E  C_ 
dom  E
6 dfrab3ss 3776 . . . . . 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 2506 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
92, 8eqtri 2496 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
109ineq1i 3696 . 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 3770 . . 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 2812 . . . . . . . 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 2825 . . . 4  |-  A. x  e.  dom  E  -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x
) )
19 rabeq0 3807 . . . 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 2496 . 2  |-  ( { x  e.  dom  E  |  N  e/  ( E `  x ) }  i^i  { x  e. 
dom  E  |  N  e.  ( E `  x
) } )  =  (/)
2210, 21eqtri 2496 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 1379    e. wcel 1767    e/ wnel 2663   A.wral 2814   {crab 2818    i^i cin 3475    C_ wss 3476   (/)c0 3785   dom cdm 4999    |` cres 5001   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-res 5011
This theorem is referenced by:  cusgrasizeinds  24152
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