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Theorem brdom 7112
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 7110 . 2  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
31, 2ax-mp 8 1  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550    e. wcel 1725   _Vcvv 2948   class class class wbr 4204   -1-1->wf1 5443    ~<_ cdom 7099
This theorem is referenced by:  domen  7113  domtr  7152  sbthlem10  7218  1sdom  7303  ac10ct  7907  domtriomlem  8314  2ndcdisj  17511  birthdaylem3  20784
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-13 1727  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  ax-un 4693
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-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-fn 5449  df-f 5450  df-f1 5451  df-dom 7103
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