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Theorem dff12 5630
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 5451 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
2 funcnv2 5502 . . 3  |-  ( Fun  `' F  <->  A. y E* x  x F y )
32anbi2i 676 . 2  |-  ( ( F : A --> B  /\  Fun  `' F )  <->  ( F : A --> B  /\  A. y E* x  x F y ) )
41, 3bitri 241 1  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. y E* x  x F y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   A.wal 1549   E*wmo 2281   class class class wbr 4204   `'ccnv 4869   Fun wfun 5440   -->wf 5442   -1-1->wf1 5443
This theorem is referenced by:  dff13  5996  fseqenlem2  7898  s4f1o  11857  2ndcdisj  17511  usgraexmpl  21412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-fun 5448  df-f1 5451
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