Theorem funop 5776

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 5329, as relsnopg 4787 is to relop 4836. (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 2206	. . 3 |- <. X , Y >. = <. X , Y >.
2		funopsn.x	. . . 4 |- X e. _V
3		funopsn.y	. . . 4 |- Y e. _V
4	2, 3	funopsn 5775	. . 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 2776	. . . . . 6 |- a e. _V
7	6, 6	funsn 5331	. . . . 5 |- Fun { <. a , a >. }
8		funeq 5300	. . . . 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 1622	. 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 1373   E.wex 1516   e. wcel 2177   _Vcvv 2773   {csn 3638   <.cop 3641   Fun wfun 5274

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-fun 5282

This theorem is referenced by:  funopdmsn  5777