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Theorem dff12 5762
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 5575 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
2 funcnv2 5629 . . 3  |-  ( Fun  `' F  <->  A. y E* x  x F y )
32anbi2i 692 . 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 367   A.wal 1396   E*wmo 2285   class class class wbr 4439   `'ccnv 4987   Fun wfun 5564   -->wf 5566   -1-1->wf1 5567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-fun 5572  df-f1 5575
This theorem is referenced by:  dff13  6141  fseqenlem2  8397  s4f1o  12857  2ndcdisj  20123  usgraexmpl  24603
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