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Theorem dff12 5786
Description: Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
Assertion
Ref Expression
dff12  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. y E* x  x F y ) )
Distinct variable group:    x, y, F
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem dff12
StepHypRef Expression
1 df-f1 5599 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
2 funcnv2 5653 . . 3  |-  ( Fun  `' F  <->  A. y E* x  x F y )
32anbi2i 694 . 2  |-  ( ( F : A --> B  /\  Fun  `' F )  <->  ( F : A --> B  /\  A. y E* x  x F y ) )
41, 3bitri 249 1  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. y E* x  x F y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   A.wal 1377   E*wmo 2276   class class class wbr 4453   `'ccnv 5004   Fun wfun 5588   -->wf 5590   -1-1->wf1 5591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-fun 5596  df-f1 5599
This theorem is referenced by:  dff13  6165  fseqenlem2  8418  s4f1o  12846  2ndcdisj  19825  usgraexmpl  24224
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