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Theorem dffo3 5853
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 5619 . 2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 ffn 5554 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
3 fnrnfv 5733 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
43eqeq1d 2446 . . . . 5  |-  ( F  Fn  A  ->  ( ran  F  =  B  <->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B ) )
52, 4syl 16 . . . 4  |-  ( F : A --> B  -> 
( ran  F  =  B 
<->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  =  B ) )
6 simpr 461 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  y  =  ( F `  x ) )
7 ffvelrn 5836 . . . . . . . . . . . 12  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
87adantr 465 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  ( F `  x )  e.  B
)
96, 8eqeltrd 2512 . . . . . . . . . 10  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  y  =  ( F `  x ) )  ->  y  e.  B )
109exp31 604 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x  e.  A  ->  ( y  =  ( F `  x )  ->  y  e.  B
) ) )
1110rexlimdv 2835 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  e.  A  y  =  ( F `  x )  ->  y  e.  B
) )
1211biantrurd 508 . . . . . . 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 628 . . . . . . 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 264 . . . . . 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 1679 . . . . 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 2544 . . . . 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 2715 . . . . 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 288 . . . 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 253 . . 3  |-  ( F : A --> B  -> 
( ran  F  =  B 
<-> 
A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
2019pm5.32i 637 . 2  |-  ( ( F : A --> B  /\  ran  F  =  B )  <-> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
211, 20bitri 249 1  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756   {cab 2424   A.wral 2710   E.wrex 2711   ran crn 4836    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   ` cfv 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421
This theorem is referenced by:  dffo4  5854  foelrn  5857  foco2  5858  fcofo  5987  foov  6232  resixpfo  7293  fofinf1o  7584  wdom2d  7787  brwdom3  7789  isf32lem9  8522  hsmexlem2  8588  cnref1o  10978  1arith  13980  orbsta  15822  symgextfo  15918  symgfixfo  15936  pwssplit1  17120  znf1o  17964  cygznlem3  17982  recosf1o  21971  efif1olem4  21981  dvdsmulf1o  22514  subfacp1lem3  27039  cvmfolem  27137  finixpnum  28385  wwlktovfo  30224  wlknwwlknsur  30315  wlkiswwlksur  30322  wwlkextsur  30334  clwwlkfo  30430  clwlkfoclwwlk  30489  frgrancvvdeqlemC  30603  numclwlk1lem2fo  30659
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