Theorem funop 5762

Description: An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 5319, as relsnopg 4778 is to relop 4827. (New usage is discouraged.)

Hypotheses

Ref	Expression
funopsn.x	|- X e. _V
funopsn.y	|- Y e. _V

Assertion

Ref	Expression
funop	|- ( Fun <. X , Y >. <-> E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) )

Distinct variable groups:   X, a   Y, a

Proof of Theorem funop

Step	Hyp	Ref	Expression
1		eqid 2204	. . 3 |- <. X , Y >. = <. X , Y >.
2		funopsn.x	. . . 4 |- X e. _V
3		funopsn.y	. . . 4 |- Y e. _V
4	2, 3	funopsn 5761	. . 3 |- ( ( Fun <. X , Y >. /\ <. X , Y >. = <. X , Y >. ) -> E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) )
5	1, 4	mpan2 425	. 2 |- ( Fun <. X , Y >. -> E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) )
6		vex 2774	. . . . . 6 |- a e. _V
7	6, 6	funsn 5321	. . . . 5 |- Fun { <. a , a >. }
8		funeq 5290	. . . . 5 |- ( <. X , Y >. = { <. a , a >. } -> ( Fun <. X , Y >. <-> Fun { <. a , a >. } ) )
9	7, 8	mpbiri 168	. . . 4 |- ( <. X , Y >. = { <. a , a >. } -> Fun <. X , Y >. )
10	9	adantl 277	. . 3 |- ( ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) -> Fun <. X , Y >. )
11	10	exlimiv 1620	. 2 |- ( E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) -> Fun <. X , Y >. )
12	5, 11	impbii 126	1 |- ( Fun <. X , Y >. <-> E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1372   E.wex 1514   e. wcel 2175   _Vcvv 2771   {csn 3632   <.cop 3635   Fun wfun 5264

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-fun 5272

This theorem is referenced by:  funopdmsn  5763