Theorem funopsn 5816

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 5366, as relsnopg 4822 is to relop 4871. (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 4319	. . 3 |- ( E. b b e. <. X , Y >. <-> ( X e. _V /\ Y e. _V ) )
4	1, 2, 3	mpbir2an 948	. 2 |- E. b b e. <. X , Y >.
5		funrel 5334	. . . . . . 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 2309	. . . . . 6 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) -> b e. F )
10		elrel 4820	. . . . . 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 2802	. . . . . . . . . . . . . . . 16 |- c e. _V
13		vex 2802	. . . . . . . . . . . . . . . 16 |- d e. _V
14	12, 13	dfop 3855	. . . . . . . . . . . . . . 15 |- <. c , d >. = { { c } , { c , d } }
15	14	eqeq2i 2240	. . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . . . 17 |- b e. _V
19	18, 1, 2	elop 4316	. . . . . . . . . . . . . . . 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 3680	. . . . . . . . . . . . . . . . . . 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 5339	. . . . . . . . . . . . . . . . . . . . . 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 5351	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( X e. _V /\ Y e. _V /\ Fun <. X , Y >. ) -> X = Y )
27	1, 2, 25, 26	mp3an12i 1375	. . . . . . . . . . . . . . . . . . . 20 |- ( ( Fun F /\ F = <. X , Y >. ) -> X = Y )
28	27	preq2d 3750	. . . . . . . . . . . . . . . . . . 19 |- ( ( Fun F /\ F = <. X , Y >. ) -> { X , X } = { X , Y } )
29	21, 28	eqtrid 2274	. . . . . . . . . . . . . . . . . 18 |- ( ( Fun F /\ F = <. X , Y >. ) -> { X } = { X , Y } )
30	29	eqeq2d 2241	. . . . . . . . . . . . . . . . 17 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( b = { X } <-> b = { X , Y } ) )
31	30	orbi2d 795	. . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . 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 2264	. . . . . . . . . . 11 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { { c } , { c , d } } = { X } )
38	12	snex 4268	. . . . . . . . . . . 12 |- { c } e. _V
39		zfpair2 4293	. . . . . . . . . . . 12 |- { c , d } e. _V
40	38, 39, 1	preqsn 3852	. . . . . . . . . . 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 2263	. . . . . . . 8 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> X = { c } )
45	1, 2	dfop 3855	. . . . . . . . . . 11 |- <. X , Y >. = { { X } , { X , Y } }
46		dfsn2 3680	. . . . . . . . . . . 12 |- { { X } } = { { X } , { X } }
47	29	preq2d 3750	. . . . . . . . . . . 12 |- ( ( Fun F /\ F = <. X , Y >. ) -> { { X } , { X } } = { { X } , { X , Y } } )
48	46, 47	eqtrid 2274	. . . . . . . . . . 11 |- ( ( Fun F /\ F = <. X , Y >. ) -> { { X } } = { { X } , { X , Y } } )
49	45, 23, 48	3eqtr4a 2288	. . . . . . . . . 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 3679	. . . . . . . . . . 11 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { X } = { { c } } )
52	51	sneqd 3679	. . . . . . . . . 10 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { { X } } = { { { c } } } )
53	12	opid 3874	. . . . . . . . . . 11 |- <. c , c >. = { { c } }
54	53	sneqi 3678	. . . . . . . . . 10 |- { <. c , c >. } = { { { c } } }
55	52, 54	eqtr4di 2280	. . . . . . . . 9 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> { { X } } = { <. c , c >. } )
56	50, 55	eqtrd 2262	. . . . . . . 8 |- ( ( ( ( Fun F /\ F = <. X , Y >. ) /\ b e. <. X , Y >. ) /\ b = <. c , d >. ) -> F = { <. c , c >. } )
57		sneq 3677	. . . . . . . . . . 11 |- ( a = c -> { a } = { c } )
58	57	eqeq2d 2241	. . . . . . . . . 10 |- ( a = c -> ( X = { a } <-> X = { c } ) )
59		id 19	. . . . . . . . . . . . 13 |- ( a = c -> a = c )
60	59, 59	opeq12d 3864	. . . . . . . . . . . 12 |- ( a = c -> <. a , a >. = <. c , c >. )
61	60	sneqd 3679	. . . . . . . . . . 11 |- ( a = c -> { <. a , a >. } = { <. c , c >. } )
62	61	eqeq2d 2241	. . . . . . . . . 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 2898	. . . . . . . 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 1944	. . . . 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 1644	. 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 713   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   {csn 3666   {cpr 3667   <.cop 3669   Rel wrel 4723   Fun wfun 5311

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-fun 5319

This theorem is referenced by:  funop  5817