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Theorem pw2f1o2val2 27111
Description: Membership in a mapped set under the pw2f1o2 27109 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 27110 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
32eleq2d 2503 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  ( Y  e.  ( F `  X )  <->  Y  e.  ( `' X " { 1o } ) ) )
43adantr 452 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
Y  e.  ( `' X " { 1o } ) ) )
5 elmapi 7038 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  X : A --> 2o )
6 ffn 5591 . . . 4  |-  ( X : A --> 2o  ->  X  Fn  A )
7 fniniseg 5851 . . . 4  |-  ( X  Fn  A  ->  ( Y  e.  ( `' X " { 1o }
)  <->  ( Y  e.  A  /\  ( X `
 Y )  =  1o ) ) )
85, 6, 73syl 19 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  ( Y  e.  ( `' X " { 1o }
)  <->  ( Y  e.  A  /\  ( X `
 Y )  =  1o ) ) )
98baibd 876 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( `' X " { 1o } )  <->  ( X `  Y )  =  1o ) )
104, 9bitrd 245 1  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
( X `  Y
)  =  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814    e. cmpt 4266   `'ccnv 4877   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   1oc1o 6717   2oc2o 6718    ^m cmap 7018
This theorem is referenced by:  wepwsolem  27116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020
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