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Theorem bren 6228
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 6227 . 2  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
2 f1ofn 5127 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f  Fn  A )
3 fndm 4998 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
4 vex 2560 . . . . . . 7  |-  f  e. 
_V
54dmex 4598 . . . . . 6  |-  dom  f  e.  _V
63, 5syl6eqelr 2129 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
72, 6syl 14 . . . 4  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
8 f1ofo 5133 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
9 forn 5109 . . . . . 6  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
108, 9syl 14 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  ran  f  =  B )
114rnex 4599 . . . . 5  |-  ran  f  e.  _V
1210, 11syl6eqelr 2129 . . . 4  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
137, 12jca 290 . . 3  |-  ( f : A -1-1-onto-> B  ->  ( A  e.  _V  /\  B  e. 
_V ) )
1413exlimiv 1489 . 2  |-  ( E. f  f : A -1-1-onto-> B  ->  ( A  e.  _V  /\  B  e.  _V )
)
15 f1oeq2 5118 . . . 4  |-  ( x  =  A  ->  (
f : x -1-1-onto-> y  <->  f : A
-1-1-onto-> y ) )
1615exbidv 1706 . . 3  |-  ( x  =  A  ->  ( E. f  f :
x
-1-1-onto-> y 
<->  E. f  f : A -1-1-onto-> y ) )
17 f1oeq3 5119 . . . 4  |-  ( y  =  B  ->  (
f : A -1-1-onto-> y  <->  f : A
-1-1-onto-> B ) )
1817exbidv 1706 . . 3  |-  ( y  =  B  ->  ( E. f  f : A
-1-1-onto-> y 
<->  E. f  f : A -1-1-onto-> B ) )
19 df-en 6222 . . 3  |-  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
2016, 18, 19brabg 4006 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  ~~  B  <->  E. f  f : A -1-1-onto-> B
) )
211, 14, 20pm5.21nii 620 1  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   _Vcvv 2557   class class class wbr 3764   dom cdm 4345   ran crn 4346    Fn wfn 4897   -onto->wfo 4900   -1-1-onto->wf1o 4901    ~~ cen 6219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-en 6222
This theorem is referenced by:  domen  6232  f1oen3g  6234  ener  6259  en0  6275  ensn1  6276  en1  6279  unen  6293  enm  6294  phplem4  6318  phplem4on  6329  fidceq  6330  dif1en  6337  fin0  6342  fin0or  6343
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