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Theorem dom3d 7149
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 7147 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
4 f1f 5639 . . . . 5  |-  ( ( x  e.  A  |->  C ) : A -1-1-> B  ->  ( x  e.  A  |->  C ) : A --> B )
53, 4syl 16 . . . 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 5603 . . . 4  |-  ( ( ( x  e.  A  |->  C ) : A --> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( x  e.  A  |->  C )  e.  _V )
95, 6, 7, 8syl3anc 1184 . . 3  |-  ( ph  ->  ( x  e.  A  |->  C )  e.  _V )
10 f1eq1 5634 . . . 4  |-  ( z  =  ( x  e.  A  |->  C )  -> 
( z : A -1-1-> B  <-> 
( x  e.  A  |->  C ) : A -1-1-> B ) )
1110spcegv 3037 . . 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 58 . 2  |-  ( ph  ->  E. z  z : A -1-1-> B )
13 brdomg 7118 . . 3  |-  ( B  e.  W  ->  ( A  ~<_  B  <->  E. z 
z : A -1-1-> B
) )
147, 13syl 16 . 2  |-  ( ph  ->  ( A  ~<_  B  <->  E. z 
z : A -1-1-> B
) )
1512, 14mpbird 224 1  |-  ( ph  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956   class class class wbr 4212    e. cmpt 4266   -->wf 5450   -1-1->wf1 5451    ~<_ cdom 7107
This theorem is referenced by:  dom3  7151  xpdom2  7203  fopwdom  7216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fv 5462  df-dom 7111
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