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Theorem f1ofveu 6339
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 5485 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1of 5472 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
31, 2syl 15 . . 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 457 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  <. C ,  x >.  e.  `' F )
6 f1ocnvfvb 5795 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
763com23 1157 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
8 dff1o4 5480 . . . . . . 7  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
98simprbi 450 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  `' F  Fn  B )
10 fnopfvb 5564 . . . . . . 7  |-  ( ( `' F  Fn  B  /\  C  e.  B
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
11103adant3 975 . . . . . 6  |-  ( ( `' F  Fn  B  /\  C  e.  B  /\  x  e.  A
)  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F
) )
129, 11syl3an1 1215 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( `' F `  C )  =  x  <->  <. C ,  x >.  e.  `' F ) )
137, 12bitrd 244 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B  /\  x  e.  A )  ->  ( ( F `  x )  =  C  <->  <. C ,  x >.  e.  `' F ) )
14133expa 1151 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  <. C ,  x >.  e.  `' F
) )
1514reubidva 2723 . 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 223 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E!wreu 2545   <.cop 3643   `'ccnv 4688    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255
This theorem is referenced by:  1arith2  12975  disjrdx  23370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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