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Theorem dom3 6638
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 9 . 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 9 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )
5 simpl 108 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
6 simpr 109 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
72, 4, 5, 6dom3d 6636 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   class class class wbr 3899    ~<_ cdom 6601
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fv 5101  df-dom 6604
This theorem is referenced by: (None)
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