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Theorem dom3 7110
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 444 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
6 simpr 448 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
72, 4, 5, 6dom3d 7108 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172    ~<_ cdom 7066
This theorem is referenced by:  canth2  7219  limenpsi  7241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fv 5421  df-dom 7070
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