Theorem fnsn 5386

Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
fnsn.1	|- A e. _V
fnsn.2	|- B e. _V

Assertion

Ref	Expression
fnsn	|- { <. A , B >. } Fn { A }

Proof of Theorem fnsn

Step	Hyp	Ref	Expression
1		fnsn.1	. 2 |- A e. _V
2		fnsn.2	. 2 |- B e. _V
3		fnsng 5379	. 2 |- ( ( A e. _V /\ B e. _V ) -> { <. A , B >. } Fn { A } )
4	1, 2, 3	mp2an 426	1 |- { <. A , B >. } Fn { A }

Syntax hints:   e. wcel 2201   _Vcvv 2801   {csn 3670   <.cop 3673   Fn wfn 5323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-fun 5330  df-fn 5331

This theorem is referenced by:  f1osn  5628  fvsnun2  5855  elixpsn  6909  xnn0nnen  10705