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Theorem fnsng 5576
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
fnsng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  Fn  { A } )

Proof of Theorem fnsng
StepHypRef Expression
1 funsng 5575 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Fun  { <. A ,  B >. } )
2 dmsnopg 5421 . . 3  |-  ( B  e.  W  ->  dom  {
<. A ,  B >. }  =  { A }
)
32adantl 466 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  dom  { <. A ,  B >. }  =  { A } )
4 df-fn 5532 . 2  |-  ( {
<. A ,  B >. }  Fn  { A }  <->  ( Fun  { <. A ,  B >. }  /\  dom  {
<. A ,  B >. }  =  { A }
) )
51, 3, 4sylanbrc 664 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  Fn  { A } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3988   <.cop 3994   dom cdm 4951   Fun wfun 5523    Fn wfn 5524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-fun 5531  df-fn 5532
This theorem is referenced by:  fnsn  5582  fnunsn  5629  fsnunfv  6030  suppsnop  6817  m1detdiag  18545  mat1dimscm  31060
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