ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bren Unicode version

Theorem bren 6259
Description: Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
Assertion
Ref Expression
bren  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
Distinct variable groups:    A, f    B, f

Proof of Theorem bren
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 encv 6258 . 2  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
2 f1ofn 5155 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f  Fn  A )
3 fndm 5026 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
4 vex 2577 . . . . . . 7  |-  f  e. 
_V
54dmex 4626 . . . . . 6  |-  dom  f  e.  _V
63, 5syl6eqelr 2145 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
72, 6syl 14 . . . 4  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
8 f1ofo 5161 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
9 forn 5137 . . . . . 6  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
108, 9syl 14 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ran  f  =  B )
114rnex 4627 . . . . 5  |-  ran  f  e.  _V
1210, 11syl6eqelr 2145 . . . 4  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
137, 12jca 294 . . 3  |-  ( f : A -1-1-onto-> B  ->  ( A  e.  _V  /\  B  e. 
_V ) )
1413exlimiv 1505 . 2  |-  ( E. f  f : A -1-1-onto-> B  ->  ( A  e.  _V  /\  B  e.  _V )
)
15 f1oeq2 5146 . . . 4  |-  ( x  =  A  ->  (
f : x -1-1-onto-> y  <->  f : A
-1-1-onto-> y ) )
1615exbidv 1722 . . 3  |-  ( x  =  A  ->  ( E. f  f :
x
-1-1-onto-> y 
<->  E. f  f : A -1-1-onto-> y ) )
17 f1oeq3 5147 . . . 4  |-  ( y  =  B  ->  (
f : A -1-1-onto-> y  <->  f : A
-1-1-onto-> B ) )
1817exbidv 1722 . . 3  |-  ( y  =  B  ->  ( E. f  f : A
-1-1-onto-> y 
<->  E. f  f : A -1-1-onto-> B ) )
19 df-en 6253 . . 3  |-  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
2016, 18, 19brabg 4034 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  ~~  B  <->  E. f  f : A -1-1-onto-> B
) )
211, 14, 20pm5.21nii 630 1  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574   class class class wbr 3792   dom cdm 4373   ran crn 4374    Fn wfn 4925   -onto->wfo 4928   -1-1-onto->wf1o 4929    ~~ cen 6250
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-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-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-en 6253
This theorem is referenced by:  domen  6263  f1oen3g  6265  ener  6290  en0  6306  ensn1  6307  en1  6310  unen  6324  enm  6325  phplem4  6349  phplem4on  6360  fidceq  6361  dif1en  6368  fin0  6373  fin0or  6374
  Copyright terms: Public domain W3C validator