MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1ofveu Structured version   Unicode version

Theorem f1ofveu 6227
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 5765 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1of 5753 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
31, 2syl 17 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' F : B --> A )
4 feu 5698 . . 3  |-  ( ( `' F : B --> A  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
53, 4sylan 469 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
6 f1ocnvfvb 6120 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
763com23 1201 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
8 dff1o4 5761 . . . . . . 7  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
98simprbi 462 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  `' F  Fn  B )
10 fnopfvb 5844 . . . . . . 7  |-  ( ( `' F  Fn  B  /\  C  e.  B
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
11103adant3 1015 . . . . . 6  |-  ( ( `' F  Fn  B  /\  C  e.  B  /\  x  e.  A
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
129, 11syl3an1 1261 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F ) )
137, 12bitrd 253 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <->  <. C ,  x >.  e.  `' F ) )
14133expa 1195 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  <. C ,  x >.  e.  `' F
) )
1514reubidva 2988 . 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 232 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  ( F `  x )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   E!wreu 2753   <.cop 3975   `'ccnv 4939    Fn wfn 5518   -->wf 5519   -1-1-onto->wf1o 5522   ` cfv 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531
This theorem is referenced by:  1arith2  14545  disjrdx  27764
  Copyright terms: Public domain W3C validator