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Theorem dff12 5604
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 5422 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
2 funcnv2 5476 . . 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 1367   E*wmo 2254   class class class wbr 4291   `'ccnv 4838   Fun wfun 5411   -->wf 5413   -1-1->wf1 5414
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 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
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 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-fun 5419  df-f1 5422
This theorem is referenced by:  dff13  5970  fseqenlem2  8194  s4f1o  12527  2ndcdisj  19059  usgraexmpl  23318
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