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Theorem cnvf1o 6233
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 2296 . 2  |-  ( x  e.  A  |->  U. `' { x } )  =  ( x  e.  A  |->  U. `' { x } )
2 snex 4232 . . . . 5  |-  { x }  e.  _V
32cnvex 5225 . . . 4  |-  `' {
x }  e.  _V
43uniex 4532 . . 3  |-  U. `' { x }  e.  _V
54a1i 10 . 2  |-  ( ( Rel  A  /\  x  e.  A )  ->  U. `' { x }  e.  _V )
6 snex 4232 . . . . 5  |-  { y }  e.  _V
76cnvex 5225 . . . 4  |-  `' {
y }  e.  _V
87uniex 4532 . . 3  |-  U. `' { y }  e.  _V
98a1i 10 . 2  |-  ( ( Rel  A  /\  y  e.  `' A )  ->  U. `' { y }  e.  _V )
10 cnvf1olem 6232 . . 3  |-  ( ( Rel  A  /\  (
x  e.  A  /\  y  =  U. `' {
x } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
11 relcnv 5067 . . . . 5  |-  Rel  `' A
12 simpr 447 . . . . 5  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
13 cnvf1olem 6232 . . . . 5  |-  ( ( Rel  `' A  /\  ( y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
1411, 12, 13sylancr 644 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
15 dfrel2 5140 . . . . . . 7  |-  ( Rel 
A  <->  `' `' A  =  A
)
16 eleq2 2357 . . . . . . 7  |-  ( `' `' A  =  A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
1715, 16sylbi 187 . . . . . 6  |-  ( Rel 
A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
1817anbi1d 685 . . . . 5  |-  ( Rel 
A  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
1918adantr 451 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
2014, 19mpbid 201 . . 3  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  A  /\  y  =  U. `' { x } ) )
2110, 20impbida 805 . 2  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  y  =  U. `' {
x } )  <->  ( y  e.  `' A  /\  x  =  U. `' { y } ) ) )
221, 5, 9, 21f1od 6083 1  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   Rel wrel 4710   -1-1-onto->wf1o 5270
This theorem is referenced by:  tposf12  6275  cnven  6952  xpcomf1o  6967  fsumcnv  12252  gsumcom2  15242
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-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-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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6138  df-2nd 6139
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