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Theorem brdom 7149
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.)
Hypothesis
Ref Expression
bren.1  |-  B  e. 
_V
Assertion
Ref Expression
brdom  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
Distinct variable groups:    A, f    B, f

Proof of Theorem brdom
StepHypRef Expression
1 bren.1 . 2  |-  B  e. 
_V
2 brdomg 7147 . 2  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
31, 2ax-mp 5 1  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    e. wcel 1727   _Vcvv 2962   class class class wbr 4237   -1-1->wf1 5480    ~<_ cdom 7136
This theorem is referenced by:  domen  7150  domtr  7189  sbthlem10  7255  1sdom  7340  ac10ct  7946  domtriomlem  8353  2ndcdisj  17550  birthdaylem3  20823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-xp 4913  df-rel 4914  df-cnv 4915  df-dm 4917  df-rn 4918  df-fn 5486  df-f 5487  df-f1 5488  df-dom 7140
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