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Theorem dom3d 6343
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 6341 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
4 f1f 5143 . . . . 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 5110 . . . 4  |-  ( ( ( x  e.  A  |->  C ) : A --> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( x  e.  A  |->  C )  e.  _V )
95, 6, 7, 8syl3anc 1170 . . 3  |-  ( ph  ->  ( x  e.  A  |->  C )  e.  _V )
10 f1eq1 5138 . . . 4  |-  ( z  =  ( x  e.  A  |->  C )  -> 
( z : A -1-1-> B  <-> 
( x  e.  A  |->  C ) : A -1-1-> B ) )
1110spcegv 2695 . . 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 6317 . . 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 1285   E.wex 1422    e. wcel 1434   _Vcvv 2610   class class class wbr 3805    |-> cmpt 3859   -->wf 4948   -1-1->wf1 4949    ~<_ cdom 6308
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fv 4960  df-dom 6311
This theorem is referenced by:  dom3  6345  xpdom2  6397  fopwdom  6402
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