Theorem funop 5830

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 5376, as relsnopg 4830 is to relop 4880. (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 2231	. . 3 |- <. X , Y >. = <. X , Y >.
2		funopsn.x	. . . 4 |- X e. _V
3		funopsn.y	. . . 4 |- Y e. _V
4	2, 3	funopsn 5829	. . 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 2805	. . . . . 6 |- a e. _V
7	6, 6	funsn 5378	. . . . 5 |- Fun { <. a , a >. }
8		funeq 5346	. . . . 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 1646	. 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 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   {csn 3669   <.cop 3672   Fun wfun 5320

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 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-fun 5328

This theorem is referenced by:  funopdmsn  5833