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

Theorem funsn 5618
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
Hypotheses
Ref Expression
funsn.1  |-  A  e. 
_V
funsn.2  |-  B  e. 
_V
Assertion
Ref Expression
funsn  |-  Fun  { <. A ,  B >. }

Proof of Theorem funsn
StepHypRef Expression
1 funsn.1 . 2  |-  A  e. 
_V
2 funsn.2 . 2  |-  B  e. 
_V
3 funsng 5616 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  Fun  { <. A ,  B >. } )
41, 2, 3mp2an 670 1  |-  Fun  { <. A ,  B >. }
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1823   _Vcvv 3106   {csn 4016   <.cop 4022   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-fun 5572
This theorem is referenced by:  funtp  5622  fun0  5627  fvsn  6080  dcomex  8818  axdc3lem4  8824  xpsc0  15049  xpsc1  15050  wfrlem13  29595  bnj1421  34499
  Copyright terms: Public domain W3C validator