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Theorem dom3 7153
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain.  C and  D can be read  C ( x ) and  D ( y ), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
Hypotheses
Ref Expression
dom2.1  |-  ( x  e.  A  ->  C  e.  B )
dom2.2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <-> 
x  =  y ) )
Assertion
Ref Expression
dom3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
Distinct variable groups:    x, y, A    x, B, y    y, C    x, D    x, V, y    x, W, y
Allowed substitution hints:    C( x)    D( y)

Proof of Theorem dom3
StepHypRef Expression
1 dom2.1 . . 3  |-  ( x  e.  A  ->  C  e.  B )
21a1i 11 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  A  ->  C  e.  B ) )
3 dom2.2 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <-> 
x  =  y ) )
43a1i 11 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )
5 simpl 445 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
6 simpr 449 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
72, 4, 5, 6dom3d 7151 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4214    ~<_ cdom 7109
This theorem is referenced by:  canth2  7262  limenpsi  7284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fv 5464  df-dom 7113
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