Theorem funopsn 5761

Description: If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) 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
funopsn	|- ( ( Fun F /\ F = <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )

Distinct variable groups:   F, a   X, a   Y, a

Proof of Theorem funopsn

Dummy variables b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopsn.x	. . 3 |- X e. _V
2		funopsn.y	. . 3 |- Y e. _V
3		opm 4277	. . 3 |- ( E. b b e. <. X , Y >. <-> ( X e. _V /\ Y e. _V ) )
4	1, 2, 3	mpbir2an 944	. 2 |- E. b b e. <. X , Y >.
5		funrel 5287	. . . . . . 7 |- ( Fun F -> Rel F )
6	5	ad2antrr 488	. . . . . 6 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> Rel F )
7		simpr 110	. . . . . . 7 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> b e. <. X , Y >. )
8		simplr 528	. . . . . . 7 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> F = <. X , Y >. )
9	7, 8	eleqtrrd 2284	. . . . . 6 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> b e. F )
10		elrel 4776	. . . . . 6 |- ( ( Rel F /\ b e. F ) -> E. c E. d b = <. c , d >. )
11	6, 9, 10	syl2anc 411	. . . . 5 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> E. c E. d b = <. c , d >. )
12		vex 2774	. . . . . . . . . . . . . . . 16 |- c e. _V
13		vex 2774	. . . . . . . . . . . . . . . 16 |- d e. _V
14	12, 13	dfop 3817	. . . . . . . . . . . . . . 15 |- <. c , d >. = { { c } , { c , d } }
15	14	eqeq2i 2215	. . . . . . . . . . . . . 14 |- ( b = <. c , d >. <-> b = { { c } , { c , d } } )
16	15	biimpi 120	. . . . . . . . . . . . 13 |- ( b = <. c , d >. -> b = { { c } , { c , d } } )
17	16	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> b = { { c } , { c , d } } )
18		vex 2774	. . . . . . . . . . . . . . . . 17 |- b e. _V
19	18, 1, 2	elop 4274	. . . . . . . . . . . . . . . 16 |- ( b e. <. X , Y >. <-> ( b = { X } \/ b = { X , Y } ) )
20	7, 19	sylib 122	. . . . . . . . . . . . . . 15 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> ( b = { X } \/ b = { X , Y } ) )
21		dfsn2 3646	. . . . . . . . . . . . . . . . . . 19 |- { X } = { X , X }
22		simpl 109	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( Fun F /\ F = <. X , Y >. ) -> Fun F )
23		simpr 110	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( Fun F /\ F = <. X , Y >. ) -> F = <. X , Y >. )
24	23	funeqd 5292	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( Fun F <-> Fun <. X , Y >. ) )
25	22, 24	mpbid 147	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( Fun F /\ F = <. X , Y >. ) -> Fun <. X , Y >. )
26		funopg 5304	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( X e. _V /\ Y e. _V /\ Fun <. X , Y >. ) -> X = Y )
27	1, 2, 25, 26	mp3an12i 1353	. . . . . . . . . . . . . . . . . . . 20 |- ( ( Fun F /\ F = <. X , Y >. ) -> X = Y )
28	27	preq2d 3716	. . . . . . . . . . . . . . . . . . 19 |- ( ( Fun F /\ F = <. X , Y >. ) -> { X , X } = { X , Y } )
29	21, 28	eqtrid 2249	. . . . . . . . . . . . . . . . . 18 |- ( ( Fun F /\ F = <. X , Y >. ) -> { X } = { X , Y } )
30	29	eqeq2d 2216	. . . . . . . . . . . . . . . . 17 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( b = { X } <-> b = { X , Y } ) )
31	30	orbi2d 791	. . . . . . . . . . . . . . . 16 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( ( b = { X } \/ b = { X } ) <-> ( b = { X } \/ b = { X , Y } ) ) )
32	31	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> ( ( b = { X } \/ b = { X } ) <-> ( b = { X } \/ b = { X , Y } ) ) )
33	20, 32	mpbird 167	. . . . . . . . . . . . . 14 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> ( b = { X } \/ b = { X } ) )
34		oridm 758	. . . . . . . . . . . . . 14 |- ( ( b = { X } \/ b = { X } ) <-> b = { X } )
35	33, 34	sylib 122	. . . . . . . . . . . . 13 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> b = { X } )
36	35	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> b = { X } )
37	17, 36	eqtr3d 2239	. . . . . . . . . . 11 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { { c } , { c , d } } = { X } )
38	12	snex 4228	. . . . . . . . . . . 12 |- { c } e. _V
39		zfpair2 4253	. . . . . . . . . . . 12 |- { c , d } e. _V
40	38, 39, 1	preqsn 3815	. . . . . . . . . . 11 |- ( { { c } , { c , d } } = { X } <-> ( { c } = { c , d } /\ { c , d } = X ) )
41	37, 40	sylib 122	. . . . . . . . . 10 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> ( { c } = { c , d } /\ { c , d } = X ) )
42	41	simpld 112	. . . . . . . . 9 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { c } = { c , d } )
43	41	simprd 114	. . . . . . . . 9 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { c , d } = X )
44	42, 43	eqtr2d 2238	. . . . . . . 8 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> X = { c } )
45	1, 2	dfop 3817	. . . . . . . . . . 11 |- <. X , Y >. = { { X } , { X , Y } }
46		dfsn2 3646	. . . . . . . . . . . 12 |- { { X } } = { { X } , { X } }
47	29	preq2d 3716	. . . . . . . . . . . 12 |- ( ( Fun F /\ F = <. X , Y >. ) -> { { X } , { X } } = { { X } , { X , Y } } )
48	46, 47	eqtrid 2249	. . . . . . . . . . 11 |- ( ( Fun F /\ F = <. X , Y >. ) -> { { X } } = { { X } , { X , Y } } )
49	45, 23, 48	3eqtr4a 2263	. . . . . . . . . 10 |- ( ( Fun F /\ F = <. X , Y >. ) -> F = { { X } } )
50	49	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> F = { { X } } )
51	44	sneqd 3645	. . . . . . . . . . 11 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { X } = { { c } } )
52	51	sneqd 3645	. . . . . . . . . 10 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { { X } } = { { { c } } } )
53	12	opid 3836	. . . . . . . . . . 11 |- <. c , c >. = { { c } }
54	53	sneqi 3644	. . . . . . . . . 10 |- { <. c , c >. } = { { { c } } }
55	52, 54	eqtr4di 2255	. . . . . . . . 9 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { { X } } = { <. c , c >. } )
56	50, 55	eqtrd 2237	. . . . . . . 8 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> F = { <. c , c >. } )
57		sneq 3643	. . . . . . . . . . 11 |- ( a = c -> { a } = { c } )
58	57	eqeq2d 2216	. . . . . . . . . 10 |- ( a = c -> ( X = { a } <-> X = { c } ) )
59		id 19	. . . . . . . . . . . . 13 |- ( a = c -> a = c )
60	59, 59	opeq12d 3826	. . . . . . . . . . . 12 |- ( a = c -> <. a , a >. = <. c , c >. )
61	60	sneqd 3645	. . . . . . . . . . 11 |- ( a = c -> { <. a , a >. } = { <. c , c >. } )
62	61	eqeq2d 2216	. . . . . . . . . 10 |- ( a = c -> ( F = { <. a , a >. } <-> F = { <. c , c >. } ) )
63	58, 62	anbi12d 473	. . . . . . . . 9 |- ( a = c -> ( ( X = { a } /\ F = { <. a , a >. } ) <-> ( X = { c } /\ F = { <. c , c >. } ) ) )
64	12, 63	spcev 2867	. . . . . . . 8 |- ( ( X = { c } /\ F = { <. c , c >. } ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )
65	44, 56, 64	syl2anc 411	. . . . . . 7 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )
66	65	ex 115	. . . . . 6 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> ( b = <. c , d >. -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) )
67	66	exlimdvv 1920	. . . . 5 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> ( E. c E. d b = <. c , d >. -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) )
68	11, 67	mpd 13	. . . 4 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )
69	68	expcom 116	. . 3 |- ( b e. <. X , Y >. -> ( ( Fun F /\ F = <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) )
70	69	exlimiv 1620	. 2 |- ( E. b b e. <. X , Y >. -> ( ( Fun F /\ F = <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) )
71	4, 70	ax-mp 5	1 |- ( ( Fun F /\ F = <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1372   E.wex 1514   e. wcel 2175   _Vcvv 2771   {csn 3632   {cpr 3633   <.cop 3635   Rel wrel 4679   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-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:  funop  5762