Theorem funop 5883

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 5422, as relsnopg 4874 is to relop 4925. (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 2238	. . 3 |- <. X , Y >. = <. X , Y >.
2		funopsn.x	. . . 4 |- X e. _V
3		funopsn.y	. . . 4 |- Y e. _V
4	2, 3	funopsn 5882	. . 3 |- ( ( Fun <. X , Y >. /\ <. X , Y >. = <. X , Y >. ) -> E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) )
5	1, 4	mpan2 429	. 2 |- ( Fun <. X , Y >. -> E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) )
6		vex 2824	. . . . . 6 |- a e. _V
7	6, 6	funsn 5424	. . . . 5 |- Fun { <. a , a >. }
8		funeq 5392	. . . . 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 1651	. 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 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   {csn 3705   <.cop 3708   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-fun 5374

This theorem is referenced by:  funopdmsn  5886