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Theorem dom3 6921
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 10 . 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 10 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )
5 simpl 443 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
6 simpr 447 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
72, 4, 5, 6dom3d 6919 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039    ~<_ cdom 6877
This theorem is referenced by:  canth2  7030  limenpsi  7052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fv 5279  df-dom 6881
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