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Theorem dffo3 5560
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 5344 . 2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 ffn 5267 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
3 fnrnfv 5461 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
43eqeq1d 2146 . . . . 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 109 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  y  =  ( F `  x ) )
7 ffvelrn 5546 . . . . . . . . . . . 12  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
87adantr 274 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  ( F `  x )  e.  B
)
96, 8eqeltrd 2214 . . . . . . . . . 10  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  y  e.  B )
109exp31 361 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x  e.  A  ->  ( y  =  ( F `  x )  ->  y  e.  B
) ) )
1110rexlimdv 2546 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  e.  A  y  =  ( F `  x )  ->  y  e.  B
) )
1211biantrurd 303 . . . . . . 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 385 . . . . . . 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 198 . . . . . 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 1796 . . . . 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 2247 . . . . 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 2419 . . . . 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 222 . . . 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 187 . . 3  |-  ( F : A --> B  -> 
( ran  F  =  B 
<-> 
A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
2019pm5.32i 449 . 2  |-  ( ( F : A --> B  /\  ran  F  =  B )  <-> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
211, 20bitri 183 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 103    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   ran crn 4535    Fn wfn 5113   -->wf 5114   -onto->wfo 5116   ` cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126
This theorem is referenced by:  dffo4  5561  foco2  5648  fcofo  5678  foov  5910  0ct  6985  ctmlemr  6986  ctm  6987  ctssdclemn0  6988  ctssdccl  6989  enumctlemm  6992  cnref1o  9433  ctiunctlemfo  11941
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