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Theorem dff12 5452
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 5276 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
2 funcnv2 5325 . . 3  |-  ( Fun  `' F  <->  A. y E* x  x F y )
32anbi2i 675 . 2  |-  ( ( F : A --> B  /\  Fun  `' F )  <->  ( F : A --> B  /\  A. y E* x  x F y ) )
41, 3bitri 240 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 176    /\ wa 358   A.wal 1530   E*wmo 2157   class class class wbr 4039   `'ccnv 4704   Fun wfun 5265   -->wf 5267   -1-1->wf1 5268
This theorem is referenced by:  dff13  5799  fseqenlem2  7668  2ndcdisj  17198  s4f1o  28225  usgraexmpl  28267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-fun 5273  df-f1 5276
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