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Theorem f1dom2g 6782
Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6784 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
f1dom2g  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  A  ~<_  B )

Proof of Theorem f1dom2g
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 f1f 5440 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 5403 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1282 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
433coml 1212 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F  e.  _V )
5 simp3 1001 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
6 f1eq1 5435 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-> B  <->  F : A -1-1-> B ) )
76spcegv 2840 . . 3  |-  ( F  e.  _V  ->  ( F : A -1-1-> B  ->  E. f  f : A -1-1-> B ) )
84, 5, 7sylc 62 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  E. f  f : A -1-1-> B )
9 brdomg 6774 . . 3  |-  ( B  e.  W  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
1093ad2ant2 1021 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
118, 10mpbird 167 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    /\ w3a 980   E.wex 1503    e. wcel 2160   _Vcvv 2752   class class class wbr 4018   -->wf 5231   -1-1->wf1 5232    ~<_ cdom 6765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-dom 6768
This theorem is referenced by:  ssdomg  6804
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