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Theorem cusgrasizeindslem2 25144
Description: Lemma 2 for cusgrasizeinds 25146. (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 4998 . . . 4  |-  dom  F  =  dom  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) } )
3 incom 3598 . . . . 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 5087 . . . . 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 3426 . . . . . 6  |-  dom  E  C_ 
dom  E
6 dfrab3ss 3694 . . . . . 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 2460 . . . 4  |-  dom  ( E  |`  { x  e. 
dom  E  |  N  e/  ( E `  x
) } )  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
92, 8eqtri 2450 . . 3  |-  dom  F  =  { x  e.  dom  E  |  N  e/  ( E `  x ) }
109ineq1i 3603 . 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 3688 . . 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 416 . . . . . . 7  |-  ( N  e.  ( E `  x )  \/  -.  N  e.  ( E `  x ) )
13 nnel 2710 . . . . . . . 8  |-  ( -.  N  e/  ( E `
 x )  <->  N  e.  ( E `  x ) )
1413orbi1i 522 . . . . . . 7  |-  ( ( -.  N  e/  ( E `  x )  \/  -.  N  e.  ( E `  x ) )  <->  ( N  e.  ( E `  x
)  \/  -.  N  e.  ( E `  x
) ) )
1512, 14mpbir 212 . . . . . 6  |-  ( -.  N  e/  ( E `
 x )  \/ 
-.  N  e.  ( E `  x ) )
16 ianor 490 . . . . . 6  |-  ( -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x )
)  <->  ( -.  N  e/  ( E `  x
)  \/  -.  N  e.  ( E `  x
) ) )
1715, 16mpbir 212 . . . . 5  |-  -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x
) )
1817rgenw 2726 . . . 4  |-  A. x  e.  dom  E  -.  ( N  e/  ( E `  x )  /\  N  e.  ( E `  x
) )
19 rabeq0 3727 . . . 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 212 . . 3  |-  { x  e.  dom  E  |  ( N  e/  ( E `
 x )  /\  N  e.  ( E `  x ) ) }  =  (/)
2111, 20eqtri 2450 . 2  |-  ( { x  e.  dom  E  |  N  e/  ( E `  x ) }  i^i  { x  e. 
dom  E  |  N  e.  ( E `  x
) } )  =  (/)
2210, 21eqtri 2450 1  |-  ( dom 
F  i^i  { x  e.  dom  E  |  N  e.  ( E `  x
) } )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    e/ wnel 2600   A.wral 2714   {crab 2718    i^i cin 3378    C_ wss 3379   (/)c0 3704   dom cdm 4796    |` cres 4798   ` cfv 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-xp 4802  df-dm 4806  df-res 4808
This theorem is referenced by:  cusgrasizeinds  25146
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