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Theorem pw2f1o2val2 27236
Description: Membership in a mapped set under the pw2f1o2 27234 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 27235 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
32eleq2d 2363 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  ( Y  e.  ( F `  X )  <->  Y  e.  ( `' X " { 1o } ) ) )
43adantr 451 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
Y  e.  ( `' X " { 1o } ) ) )
5 elmapi 6808 . . . 4  |-  ( X  e.  ( 2o  ^m  A )  ->  X : A --> 2o )
6 ffn 5405 . . . 4  |-  ( X : A --> 2o  ->  X  Fn  A )
7 fniniseg 5662 . . . 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 875 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  Y  e.  A )  ->  ( Y  e.  ( `' X " { 1o } )  <->  ( X `  Y )  =  1o ) )
104, 9bitrd 244 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1oc1o 6488   2oc2o 6489    ^m cmap 6788
This theorem is referenced by:  wepwsolem  27241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790
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