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Theorem cnvf1o 6671
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 2443 . 2  |-  ( x  e.  A  |->  U. `' { x } )  =  ( x  e.  A  |->  U. `' { x } )
2 snex 4533 . . . . 5  |-  { x }  e.  _V
32cnvex 6525 . . . 4  |-  `' {
x }  e.  _V
43uniex 6376 . . 3  |-  U. `' { x }  e.  _V
54a1i 11 . 2  |-  ( ( Rel  A  /\  x  e.  A )  ->  U. `' { x }  e.  _V )
6 snex 4533 . . . . 5  |-  { y }  e.  _V
76cnvex 6525 . . . 4  |-  `' {
y }  e.  _V
87uniex 6376 . . 3  |-  U. `' { y }  e.  _V
98a1i 11 . 2  |-  ( ( Rel  A  /\  y  e.  `' A )  ->  U. `' { y }  e.  _V )
10 cnvf1olem 6670 . . 3  |-  ( ( Rel  A  /\  (
x  e.  A  /\  y  =  U. `' {
x } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
11 relcnv 5206 . . . . 5  |-  Rel  `' A
12 simpr 461 . . . . 5  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
13 cnvf1olem 6670 . . . . 5  |-  ( ( Rel  `' A  /\  ( y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
1411, 12, 13sylancr 663 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
15 dfrel2 5288 . . . . . . 7  |-  ( Rel 
A  <->  `' `' A  =  A
)
16 eleq2 2504 . . . . . . 7  |-  ( `' `' A  =  A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
1715, 16sylbi 195 . . . . . 6  |-  ( Rel 
A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
1817anbi1d 704 . . . . 5  |-  ( Rel 
A  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
1918adantr 465 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
2014, 19mpbid 210 . . 3  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  A  /\  y  =  U. `' { x } ) )
2110, 20impbida 828 . 2  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  y  =  U. `' {
x } )  <->  ( y  e.  `' A  /\  x  =  U. `' { y } ) ) )
221, 5, 9, 21f1od 6310 1  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972   {csn 3877   U.cuni 4091    e. cmpt 4350   `'ccnv 4839   Rel wrel 4845   -1-1-onto->wf1o 5417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-1st 6577  df-2nd 6578
This theorem is referenced by:  tposf12  6770  cnven  7385  xpcomf1o  7400  fsumcnv  13240  gsumcom2  16467  fprodcnv  27494
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