ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1ofveu Unicode version

Theorem f1ofveu 5885
Description: There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
Assertion
Ref Expression
f1ofveu  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  ( F `  x )  =  C )
Distinct variable groups:    x, A    x, B    x, C    x, F

Proof of Theorem f1ofveu
StepHypRef Expression
1 f1ocnv 5493 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1of 5480 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
31, 2syl 14 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' F : B --> A )
4 feu 5417 . . 3  |-  ( ( `' F : B --> A  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
53, 4sylan 283 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
6 f1ocnvfvb 5802 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
763com23 1211 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
8 dff1o4 5488 . . . . . . 7  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
98simprbi 275 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  `' F  Fn  B )
10 fnopfvb 5578 . . . . . . 7  |-  ( ( `' F  Fn  B  /\  C  e.  B
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
11103adant3 1019 . . . . . 6  |-  ( ( `' F  Fn  B  /\  C  e.  B  /\  x  e.  A
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
129, 11syl3an1 1282 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F ) )
137, 12bitrd 188 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <->  <. C ,  x >.  e.  `' F ) )
14133expa 1205 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  <. C ,  x >.  e.  `' F
) )
1514reubidva 2673 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( E! x  e.  A  ( F `  x )  =  C  <-> 
E! x  e.  A  <. C ,  x >.  e.  `' F ) )
165, 15mpbird 167 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  ( F `  x )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   E!wreu 2470   <.cop 3610   `'ccnv 4643    Fn wfn 5230   -->wf 5231   -1-1-onto->wf1o 5234   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243
This theorem is referenced by:  1arith2  12403
  Copyright terms: Public domain W3C validator