Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pw2f1o2val2 Structured version   Unicode version

Theorem pw2f1o2val2 35344
Description: Membership in a mapped set under the pw2f1o2 35342 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 35343 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
32eleq2d 2472 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  ( Y  e.  ( F `  X )  <->  Y  e.  ( `' X " { 1o } ) ) )
43adantr 463 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
Y  e.  ( `' X " { 1o } ) ) )
5 elmapi 7478 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  X : A --> 2o )
6 ffn 5714 . . . 4  |-  ( X : A --> 2o  ->  X  Fn  A )
7 fniniseg 5986 . . . 4  |-  ( X  Fn  A  ->  ( Y  e.  ( `' X " { 1o }
)  <->  ( Y  e.  A  /\  ( X `
 Y )  =  1o ) ) )
85, 6, 73syl 18 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  ( Y  e.  ( `' X " { 1o }
)  <->  ( Y  e.  A  /\  ( X `
 Y )  =  1o ) ) )
98baibd 910 . 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 367    = wceq 1405    e. wcel 1842   {csn 3972    |-> cmpt 4453   `'ccnv 4822   "cima 4826    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   1oc1o 7160   2oc2o 7161    ^m cmap 7457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459
This theorem is referenced by:  wepwsolem  35349
  Copyright terms: Public domain W3C validator