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Theorem dom3 6905
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 6903 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023    ~<_ cdom 6861
This theorem is referenced by:  canth2  7014  limenpsi  7036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fv 5263  df-dom 6865
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