Theorem fnsnb 6432

Description: A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)

Hypothesis

Ref	Expression
fnsnb.1	|- A e. _V

Assertion

Ref	Expression
fnsnb	|- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )

Proof of Theorem fnsnb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnresdm 6000	. . . . . . . 8 |- ( F Fn { A } -> ( F |` { A } ) = F )
2		fnfun 5988	. . . . . . . . 9 |- ( F Fn { A } -> Fun F )
3		funressn 6426	. . . . . . . . 9 |- ( Fun F -> ( F |` { A } ) C_ { <. A , ( F ` A ) >. } )
4	2, 3	syl 17	. . . . . . . 8 |- ( F Fn { A } -> ( F |` { A } ) C_ { <. A , ( F ` A ) >. } )
5	1, 4	eqsstr3d 3640	. . . . . . 7 |- ( F Fn { A } -> F C_ { <. A , ( F ` A ) >. } )
6	5	sseld 3602	. . . . . 6 |- ( F Fn { A } -> ( x e. F -> x e. { <. A , ( F ` A ) >. } ) )
7		elsni 4194	. . . . . 6 |- ( x e. { <. A , ( F ` A ) >. } -> x = <. A , ( F ` A ) >. )
8	6, 7	syl6 35	. . . . 5 |- ( F Fn { A } -> ( x e. F -> x = <. A , ( F ` A ) >. ) )
9		df-fn 5891	. . . . . . . 8 |- ( F Fn { A } <-> ( Fun F /\ dom F = { A } ) )
10		fnsnb.1	. . . . . . . . . . 11 |- A e. _V
11	10	snid 4208	. . . . . . . . . 10 |- A e. { A }
12		eleq2 2690	. . . . . . . . . 10 |- ( dom F = { A } -> ( A e. dom F <-> A e. { A } ) )
13	11, 12	mpbiri 248	. . . . . . . . 9 |- ( dom F = { A } -> A e. dom F )
14	13	anim2i 593	. . . . . . . 8 |- ( ( Fun F /\ dom F = { A } ) -> ( Fun F /\ A e. dom F ) )
15	9, 14	sylbi 207	. . . . . . 7 |- ( F Fn { A } -> ( Fun F /\ A e. dom F ) )
16		funfvop 6329	. . . . . . 7 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
17	15, 16	syl 17	. . . . . 6 |- ( F Fn { A } -> <. A , ( F ` A ) >. e. F )
18		eleq1 2689	. . . . . 6 |- ( x = <. A , ( F ` A ) >. -> ( x e. F <-> <. A , ( F ` A ) >. e. F ) )
19	17, 18	syl5ibrcom 237	. . . . 5 |- ( F Fn { A } -> ( x = <. A , ( F ` A ) >. -> x e. F ) )
20	8, 19	impbid 202	. . . 4 |- ( F Fn { A } -> ( x e. F <-> x = <. A , ( F ` A ) >. ) )
21		velsn 4193	. . . 4 |- ( x e. { <. A , ( F ` A ) >. } <-> x = <. A , ( F ` A ) >. )
22	20, 21	syl6bbr 278	. . 3 |- ( F Fn { A } -> ( x e. F <-> x e. { <. A , ( F ` A ) >. } ) )
23	22	eqrdv 2620	. 2 |- ( F Fn { A } -> F = { <. A , ( F ` A ) >. } )
24		fvex 6201	. . . 4 |- ( F ` A ) e. _V
25	10, 24	fnsn 5946	. . 3 |- { <. A , ( F ` A ) >. } Fn { A }
26		fneq1 5979	. . 3 |- ( F = { <. A , ( F ` A ) >. } -> ( F Fn { A } <-> { <. A , ( F ` A ) >. } Fn { A } ) )
27	25, 26	mpbiri 248	. 2 |- ( F = { <. A , ( F ` A ) >. } -> F Fn { A } )
28	23, 27	impbii 199	1 |- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  fnprb  6472