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Theorem dom3d 6481
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 6479 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
4 f1f 5210 . . . . 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 5173 . . . 4  |-  ( ( ( x  e.  A  |->  C ) : A --> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( x  e.  A  |->  C )  e.  _V )
95, 6, 7, 8syl3anc 1174 . . 3  |-  ( ph  ->  ( x  e.  A  |->  C )  e.  _V )
10 f1eq1 5205 . . . 4  |-  ( z  =  ( x  e.  A  |->  C )  -> 
( z : A -1-1-> B  <-> 
( x  e.  A  |->  C ) : A -1-1-> B ) )
1110spcegv 2707 . . 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 61 . 2  |-  ( ph  ->  E. z  z : A -1-1-> B )
13 brdomg 6455 . . 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 165 1  |-  ( ph  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   class class class wbr 3843    |-> cmpt 3897   -->wf 5006   -1-1->wf1 5007    ~<_ cdom 6446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fv 5018  df-dom 6449
This theorem is referenced by:  dom3  6483  xpdom2  6537  fopwdom  6542
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