ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnsng Unicode version

Theorem fnsng 5140
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 5139 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Fun  { <. A ,  B >. } )
2 dmsnopg 4980 . . 3  |-  ( B  e.  W  ->  dom  {
<. A ,  B >. }  =  { A }
)
32adantl 275 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  dom  { <. A ,  B >. }  =  { A } )
4 df-fn 5096 . 2  |-  ( {
<. A ,  B >. }  Fn  { A }  <->  ( Fun  { <. A ,  B >. }  /\  dom  {
<. A ,  B >. }  =  { A }
) )
51, 3, 4sylanbrc 413 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  Fn  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   {csn 3497   <.cop 3500   dom cdm 4509   Fun wfun 5087    Fn wfn 5088
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-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
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-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-fun 5095  df-fn 5096
This theorem is referenced by:  fnsn  5147  fnunsn  5200  fsnunfv  5589  tfr0dm  6187  ennnfonelemhom  11839
  Copyright terms: Public domain W3C validator