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Theorem dom3d 6795
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
Hypotheses
Ref Expression
dom2d.1  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
dom2d.2  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )
dom3d.3  |-  ( ph  ->  A  e.  V )
dom3d.4  |-  ( ph  ->  B  e.  W )
Assertion
Ref Expression
dom3d  |-  ( ph  ->  A  ~<_  B )
Distinct variable groups:    x, y, A   
x, B, y    y, C    x, D    ph, x, y
Allowed substitution hints:    C( x)    D( y)    V( x, y)    W( x, y)

Proof of Theorem dom3d
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dom2d.1 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
2 dom2d.2 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )
31, 2dom2lem 6793 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
4 f1f 5437 . . . . 5  |-  ( ( x  e.  A  |->  C ) : A -1-1-> B  ->  ( x  e.  A  |->  C ) : A --> B )
53, 4syl 14 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> B )
6 dom3d.3 . . . 4  |-  ( ph  ->  A  e.  V )
7 dom3d.4 . . . 4  |-  ( ph  ->  B  e.  W )
8 fex2 5400 . . . 4  |-  ( ( ( x  e.  A  |->  C ) : A --> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( x  e.  A  |->  C )  e.  _V )
95, 6, 7, 8syl3anc 1249 . . 3  |-  ( ph  ->  ( x  e.  A  |->  C )  e.  _V )
10 f1eq1 5432 . . . 4  |-  ( z  =  ( x  e.  A  |->  C )  -> 
( z : A -1-1-> B  <-> 
( x  e.  A  |->  C ) : A -1-1-> B ) )
1110spcegv 2840 . . 3  |-  ( ( x  e.  A  |->  C )  e.  _V  ->  ( ( x  e.  A  |->  C ) : A -1-1-> B  ->  E. z  z : A -1-1-> B ) )
129, 3, 11sylc 62 . 2  |-  ( ph  ->  E. z  z : A -1-1-> B )
13 brdomg 6769 . . 3  |-  ( B  e.  W  ->  ( A  ~<_  B  <->  E. z 
z : A -1-1-> B
) )
147, 13syl 14 . 2  |-  ( ph  ->  ( A  ~<_  B  <->  E. z 
z : A -1-1-> B
) )
1512, 14mpbird 167 1  |-  ( ph  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   _Vcvv 2752   class class class wbr 4018    |-> cmpt 4079   -->wf 5228   -1-1->wf1 5229    ~<_ cdom 6760
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 4189  ax-pr 4224  ax-un 4448
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-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  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-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fv 5240  df-dom 6763
This theorem is referenced by:  dom3  6797  xpdom2  6852  fopwdom  6859
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