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Theorem dom3d 6748
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 6746 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
4 f1f 5401 . . . . 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 5364 . . . 4  |-  ( ( ( x  e.  A  |->  C ) : A --> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( x  e.  A  |->  C )  e.  _V )
95, 6, 7, 8syl3anc 1233 . . 3  |-  ( ph  ->  ( x  e.  A  |->  C )  e.  _V )
10 f1eq1 5396 . . . 4  |-  ( z  =  ( x  e.  A  |->  C )  -> 
( z : A -1-1-> B  <-> 
( x  e.  A  |->  C ) : A -1-1-> B ) )
1110spcegv 2818 . . 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 6722 . . 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 166 1  |-  ( ph  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   class class class wbr 3987    |-> cmpt 4048   -->wf 5192   -1-1->wf1 5193    ~<_ cdom 6713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fv 5204  df-dom 6716
This theorem is referenced by:  dom3  6750  xpdom2  6805  fopwdom  6810
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