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Theorem f1ofveu 5632
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 5260 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1of 5247 . . . 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 5187 . . 3  |-  ( ( `' F : B --> A  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
53, 4sylan 277 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
6 f1ocnvfvb 5551 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
763com23 1149 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
8 dff1o4 5255 . . . . . . 7  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
98simprbi 269 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  `' F  Fn  B )
10 fnopfvb 5340 . . . . . . 7  |-  ( ( `' F  Fn  B  /\  C  e.  B
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
11103adant3 963 . . . . . 6  |-  ( ( `' F  Fn  B  /\  C  e.  B  /\  x  e.  A
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
129, 11syl3an1 1207 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F ) )
137, 12bitrd 186 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <->  <. C ,  x >.  e.  `' F ) )
14133expa 1143 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  <. C ,  x >.  e.  `' F
) )
1514reubidva 2549 . 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 165 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E!wreu 2361   <.cop 3447   `'ccnv 4435    Fn wfn 5005   -->wf 5006   -1-1-onto->wf1o 5009   ` cfv 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018
This theorem is referenced by: (None)
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