Theorem funop 5866

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 5407, as relsnopg 4859 is to relop 4910. (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 2234	. . 3 |- <. X , Y >. = <. X , Y >.
2		funopsn.x	. . . 4 |- X e. _V
3		funopsn.y	. . . 4 |- Y e. _V
4	2, 3	funopsn 5865	. . 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 2818	. . . . . 6 |- a e. _V
7	6, 6	funsn 5409	. . . . 5 |- Fun { <. a , a >. }
8		funeq 5377	. . . . 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 1647	. 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 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   {csn 3694   <.cop 3697   Fun wfun 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-fun 5359

This theorem is referenced by:  funopdmsn  5869