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Theorem pw2f1o2val2 29314
Description: Membership in a mapped set under the pw2f1o2 29312 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 29313 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
32eleq2d 2508 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  ( Y  e.  ( F `  X )  <->  Y  e.  ( `' X " { 1o } ) ) )
43adantr 462 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
Y  e.  ( `' X " { 1o } ) ) )
5 elmapi 7230 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  X : A --> 2o )
6 ffn 5556 . . . 4  |-  ( X : A --> 2o  ->  X  Fn  A )
7 fniniseg 5821 . . . 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 895 . 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 1364    e. wcel 1761   {csn 3874    e. cmpt 4347   `'ccnv 4835   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   1oc1o 6909   2oc2o 6910    ^m cmap 7210
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212
This theorem is referenced by:  wepwsolem  29319
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