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Theorem dffo3 5446
Description: An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
Assertion
Ref Expression
dffo3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
Distinct variable groups:    x, y, A   
x, B, y    x, F, y

Proof of Theorem dffo3
StepHypRef Expression
1 dffo2 5237 . 2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 ffn 5161 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
3 fnrnfv 5351 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
43eqeq1d 2096 . . . . 5  |-  ( F  Fn  A  ->  ( ran  F  =  B  <->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B ) )
52, 4syl 14 . . . 4  |-  ( F : A --> B  -> 
( ran  F  =  B 
<->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B ) )
6 simpr 108 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  y  =  ( F `  x ) )
7 ffvelrn 5432 . . . . . . . . . . . 12  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
87adantr 270 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  ( F `  x )  e.  B
)
96, 8eqeltrd 2164 . . . . . . . . . 10  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  y  e.  B )
109exp31 356 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x  e.  A  ->  ( y  =  ( F `  x )  ->  y  e.  B
) ) )
1110rexlimdv 2488 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  e.  A  y  =  ( F `  x )  ->  y  e.  B
) )
1211biantrurd 299 . . . . . . 7  |-  ( F : A --> B  -> 
( ( y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) )  <->  ( ( E. x  e.  A  y  =  ( F `  x )  ->  y  e.  B )  /\  (
y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) ) ) )
13 dfbi2 380 . . . . . . 7  |-  ( ( E. x  e.  A  y  =  ( F `  x )  <->  y  e.  B )  <->  ( ( E. x  e.  A  y  =  ( F `  x )  ->  y  e.  B )  /\  (
y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) ) )
1412, 13syl6rbbr 197 . . . . . 6  |-  ( F : A --> B  -> 
( ( E. x  e.  A  y  =  ( F `  x )  <-> 
y  e.  B )  <-> 
( y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) ) )
1514albidv 1752 . . . . 5  |-  ( F : A --> B  -> 
( A. y ( E. x  e.  A  y  =  ( F `  x )  <->  y  e.  B )  <->  A. y
( y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) ) )
16 abeq1 2197 . . . . 5  |-  ( { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B  <->  A. y
( E. x  e.  A  y  =  ( F `  x )  <-> 
y  e.  B ) )
17 df-ral 2364 . . . . 5  |-  ( A. y  e.  B  E. x  e.  A  y  =  ( F `  x )  <->  A. y
( y  e.  B  ->  E. x  e.  A  y  =  ( F `  x ) ) )
1815, 16, 173bitr4g 221 . . . 4  |-  ( F : A --> B  -> 
( { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B 
<-> 
A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
195, 18bitrd 186 . . 3  |-  ( F : A --> B  -> 
( ran  F  =  B 
<-> 
A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
2019pm5.32i 442 . 2  |-  ( ( F : A --> B  /\  ran  F  =  B )  <-> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
211, 20bitri 182 1  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   ran crn 4439    Fn wfn 5010   -->wf 5011   -onto->wfo 5013   ` cfv 5015
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 3957  ax-pow 4009  ax-pr 4036
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-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023
This theorem is referenced by:  dffo4  5447  foelrnOLD  5532  foco2  5533  fcofo  5563  foov  5791  cnref1o  9133
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