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Theorem pw2f1o2val2 30575
Description: Membership in a mapped set under the pw2f1o2 30573 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypothesis
Ref Expression
pw2f1o2.f  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
Assertion
Ref Expression
pw2f1o2val2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
( X `  Y
)  =  1o ) )
Distinct variable groups:    x, A    x, X    x, Y
Allowed substitution hint:    F( x)

Proof of Theorem pw2f1o2val2
StepHypRef Expression
1 pw2f1o2.f . . . . 5  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
21pw2f1o2val 30574 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
32eleq2d 2530 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  ( Y  e.  ( F `  X )  <->  Y  e.  ( `' X " { 1o } ) ) )
43adantr 465 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
Y  e.  ( `' X " { 1o } ) ) )
5 elmapi 7430 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  X : A --> 2o )
6 ffn 5722 . . . 4  |-  ( X : A --> 2o  ->  X  Fn  A )
7 fniniseg 5993 . . . 4  |-  ( X  Fn  A  ->  ( Y  e.  ( `' X " { 1o }
)  <->  ( Y  e.  A  /\  ( X `
 Y )  =  1o ) ) )
85, 6, 73syl 20 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  ( Y  e.  ( `' X " { 1o }
)  <->  ( Y  e.  A  /\  ( X `
 Y )  =  1o ) ) )
98baibd 902 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( `' X " { 1o } )  <->  ( X `  Y )  =  1o ) )
104, 9bitrd 253 1  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
( X `  Y
)  =  1o ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {csn 4020    |-> cmpt 4498   `'ccnv 4991   "cima 4995    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   1oc1o 7113   2oc2o 7114    ^m cmap 7410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412
This theorem is referenced by:  wepwsolem  30580
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