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Theorem cnvf1o 6249
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 2189 . 2  |-  ( x  e.  A  |->  U. `' { x } )  =  ( x  e.  A  |->  U. `' { x } )
2 snexg 4202 . . . 4  |-  ( x  e.  A  ->  { x }  e.  _V )
3 cnvexg 5184 . . . 4  |-  ( { x }  e.  _V  ->  `' { x }  e.  _V )
4 uniexg 4457 . . . 4  |-  ( `' { x }  e.  _V  ->  U. `' { x }  e.  _V )
52, 3, 43syl 17 . . 3  |-  ( x  e.  A  ->  U. `' { x }  e.  _V )
65adantl 277 . 2  |-  ( ( Rel  A  /\  x  e.  A )  ->  U. `' { x }  e.  _V )
7 snexg 4202 . . . 4  |-  ( y  e.  `' A  ->  { y }  e.  _V )
8 cnvexg 5184 . . . 4  |-  ( { y }  e.  _V  ->  `' { y }  e.  _V )
9 uniexg 4457 . . . 4  |-  ( `' { y }  e.  _V  ->  U. `' { y }  e.  _V )
107, 8, 93syl 17 . . 3  |-  ( y  e.  `' A  ->  U. `' { y }  e.  _V )
1110adantl 277 . 2  |-  ( ( Rel  A  /\  y  e.  `' A )  ->  U. `' { y }  e.  _V )
12 cnvf1olem 6248 . . 3  |-  ( ( Rel  A  /\  (
x  e.  A  /\  y  =  U. `' {
x } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
13 relcnv 5024 . . . . 5  |-  Rel  `' A
14 simpr 110 . . . . 5  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
15 cnvf1olem 6248 . . . . 5  |-  ( ( Rel  `' A  /\  ( y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
1613, 14, 15sylancr 414 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
17 dfrel2 5097 . . . . . . 7  |-  ( Rel 
A  <->  `' `' A  =  A
)
18 eleq2 2253 . . . . . . 7  |-  ( `' `' A  =  A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
1917, 18sylbi 121 . . . . . 6  |-  ( Rel 
A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
2019anbi1d 465 . . . . 5  |-  ( Rel 
A  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
2120adantr 276 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
2216, 21mpbid 147 . . 3  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  A  /\  y  =  U. `' { x } ) )
2312, 22impbida 596 . 2  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  y  =  U. `' {
x } )  <->  ( y  e.  `' A  /\  x  =  U. `' { y } ) ) )
241, 6, 11, 23f1od 6096 1  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   _Vcvv 2752   {csn 3607   U.cuni 3824    |-> cmpt 4079   `'ccnv 4643   Rel wrel 4649   -1-1-onto->wf1o 5234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6164  df-2nd 6165
This theorem is referenced by:  tposf12  6293  cnven  6833  xpcomf1o  6850  fsumcnv  11476  fprodcnv  11664
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