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Theorem cnvf1o 5874
Description: Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
cnvf1o  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
Distinct variable group:    x, A

Proof of Theorem cnvf1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2056 . 2  |-  ( x  e.  A  |->  U. `' { x } )  =  ( x  e.  A  |->  U. `' { x } )
2 snexg 3964 . . . 4  |-  ( x  e.  A  ->  { x }  e.  _V )
3 cnvexg 4883 . . . 4  |-  ( { x }  e.  _V  ->  `' { x }  e.  _V )
4 uniexg 4203 . . . 4  |-  ( `' { x }  e.  _V  ->  U. `' { x }  e.  _V )
52, 3, 43syl 17 . . 3  |-  ( x  e.  A  ->  U. `' { x }  e.  _V )
65adantl 266 . 2  |-  ( ( Rel  A  /\  x  e.  A )  ->  U. `' { x }  e.  _V )
7 snexg 3964 . . . 4  |-  ( y  e.  `' A  ->  { y }  e.  _V )
8 cnvexg 4883 . . . 4  |-  ( { y }  e.  _V  ->  `' { y }  e.  _V )
9 uniexg 4203 . . . 4  |-  ( `' { y }  e.  _V  ->  U. `' { y }  e.  _V )
107, 8, 93syl 17 . . 3  |-  ( y  e.  `' A  ->  U. `' { y }  e.  _V )
1110adantl 266 . 2  |-  ( ( Rel  A  /\  y  e.  `' A )  ->  U. `' { y }  e.  _V )
12 cnvf1olem 5873 . . 3  |-  ( ( Rel  A  /\  (
x  e.  A  /\  y  =  U. `' {
x } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
13 relcnv 4731 . . . . 5  |-  Rel  `' A
14 simpr 107 . . . . 5  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
15 cnvf1olem 5873 . . . . 5  |-  ( ( Rel  `' A  /\  ( y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
1613, 14, 15sylancr 399 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
17 dfrel2 4799 . . . . . . 7  |-  ( Rel 
A  <->  `' `' A  =  A
)
18 eleq2 2117 . . . . . . 7  |-  ( `' `' A  =  A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
1917, 18sylbi 118 . . . . . 6  |-  ( Rel 
A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
2019anbi1d 446 . . . . 5  |-  ( Rel 
A  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
2120adantr 265 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
2216, 21mpbid 139 . . 3  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  A  /\  y  =  U. `' { x } ) )
2312, 22impbida 538 . 2  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  y  =  U. `' {
x } )  <->  ( y  e.  `' A  /\  x  =  U. `' { y } ) ) )
241, 6, 11, 23f1od 5731 1  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   {csn 3403   U.cuni 3608    |-> cmpt 3846   `'ccnv 4372   Rel wrel 4378   -1-1-onto->wf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by:  tposf12  5915  cnven  6319  xpcomf1o  6330
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