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

Theorem f1ofveu 6276
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 5818 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1of 5806 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
31, 2syl 16 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' F : B --> A )
4 feu 5751 . . 3  |-  ( ( `' F : B --> A  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
53, 4sylan 471 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
6 f1ocnvfvb 6170 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
763com23 1203 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
8 dff1o4 5814 . . . . . . 7  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
98simprbi 464 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  `' F  Fn  B )
10 fnopfvb 5899 . . . . . . 7  |-  ( ( `' F  Fn  B  /\  C  e.  B
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
11103adant3 1017 . . . . . 6  |-  ( ( `' F  Fn  B  /\  C  e.  B  /\  x  e.  A
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
129, 11syl3an1 1262 . . . . 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 1197 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  <. C ,  x >.  e.  `' F
) )
1514reubidva 3027 . 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 369    /\ w3a 974    = wceq 1383    e. wcel 1804   E!wreu 2795   <.cop 4020   `'ccnv 4988    Fn wfn 5573   -->wf 5574   -1-1-onto->wf1o 5577   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586
This theorem is referenced by:  1arith2  14427  disjrdx  27426
  Copyright terms: Public domain W3C validator