MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvf1o Structured version   Unicode version

Theorem cnvf1o 6447
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 2438 . 2  |-  ( x  e.  A  |->  U. `' { x } )  =  ( x  e.  A  |->  U. `' { x } )
2 snex 4407 . . . . 5  |-  { x }  e.  _V
32cnvex 5408 . . . 4  |-  `' {
x }  e.  _V
43uniex 4707 . . 3  |-  U. `' { x }  e.  _V
54a1i 11 . 2  |-  ( ( Rel  A  /\  x  e.  A )  ->  U. `' { x }  e.  _V )
6 snex 4407 . . . . 5  |-  { y }  e.  _V
76cnvex 5408 . . . 4  |-  `' {
y }  e.  _V
87uniex 4707 . . 3  |-  U. `' { y }  e.  _V
98a1i 11 . 2  |-  ( ( Rel  A  /\  y  e.  `' A )  ->  U. `' { y }  e.  _V )
10 cnvf1olem 6446 . . 3  |-  ( ( Rel  A  /\  (
x  e.  A  /\  y  =  U. `' {
x } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
11 relcnv 5244 . . . . 5  |-  Rel  `' A
12 simpr 449 . . . . 5  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( y  e.  `' A  /\  x  =  U. `' { y } ) )
13 cnvf1olem 6446 . . . . 5  |-  ( ( Rel  `' A  /\  ( y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
1411, 12, 13sylancr 646 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  `' `' A  /\  y  =  U. `' { x } ) )
15 dfrel2 5323 . . . . . . 7  |-  ( Rel 
A  <->  `' `' A  =  A
)
16 eleq2 2499 . . . . . . 7  |-  ( `' `' A  =  A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
1715, 16sylbi 189 . . . . . 6  |-  ( Rel 
A  ->  ( x  e.  `' `' A  <->  x  e.  A
) )
1817anbi1d 687 . . . . 5  |-  ( Rel 
A  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
1918adantr 453 . . . 4  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( (
x  e.  `' `' A  /\  y  =  U. `' { x } )  <-> 
( x  e.  A  /\  y  =  U. `' { x } ) ) )
2014, 19mpbid 203 . . 3  |-  ( ( Rel  A  /\  (
y  e.  `' A  /\  x  =  U. `' { y } ) )  ->  ( x  e.  A  /\  y  =  U. `' { x } ) )
2110, 20impbida 807 . 2  |-  ( Rel 
A  ->  ( (
x  e.  A  /\  y  =  U. `' {
x } )  <->  ( y  e.  `' A  /\  x  =  U. `' { y } ) ) )
221, 5, 9, 21f1od 6296 1  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   U.cuni 4017    e. cmpt 4268   `'ccnv 4879   Rel wrel 4885   -1-1-onto->wf1o 5455
This theorem is referenced by:  tposf12  6506  cnven  7184  xpcomf1o  7199  fsumcnv  12559  gsumcom2  15551  fprodcnv  25309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1st 6351  df-2nd 6352
  Copyright terms: Public domain W3C validator