MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funsng Structured version   Unicode version

Theorem funsng 5632
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
Assertion
Ref Expression
funsng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Fun  { <. A ,  B >. } )

Proof of Theorem funsng
StepHypRef Expression
1 funcnvsn 5631 . 2  |-  Fun  `' { <. B ,  A >. }
2 cnvsng 5492 . . . 4  |-  ( ( B  e.  W  /\  A  e.  V )  ->  `' { <. B ,  A >. }  =  { <. A ,  B >. } )
32ancoms 453 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' { <. B ,  A >. }  =  { <. A ,  B >. } )
43funeqd 5607 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( Fun  `' { <. B ,  A >. }  <->  Fun  { <. A ,  B >. } ) )
51, 4mpbii 211 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Fun  { <. A ,  B >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4027   <.cop 4033   `'ccnv 4998   Fun wfun 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-fun 5588
This theorem is referenced by:  fnsng  5633  funsn  5634  funprg  5635  funtpg  5636  tfrlem10  7053  snopfsupp  7848  funsnfsupp  7849  strle1  14582  constr3pthlem1  24331  bnj519  32871  bnj150  33013
  Copyright terms: Public domain W3C validator