Theorem fvsingle 30735

Description: The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)

Assertion

Ref	Expression
fvsingle	|- (Singleton ` A ) = { A }

Proof of Theorem fvsingle

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5887	. . . 4 |- ( x = A -> (Singleton ` x ) = (Singleton ` A ) )
2		sneq 3989	. . . 4 |- ( x = A -> { x } = { A } )
3	1, 2	eqeq12d 2476	. . 3 |- ( x = A -> ( (Singleton ` x ) = { x } <-> (Singleton ` A ) = { A } ) )
4		eqid 2461	. . . . 5 |- { x } = { x }
5		vex 3059	. . . . . 6 |- x e. _V
6		snex 4654	. . . . . 6 |- { x } e. _V
7	5, 6	brsingle 30732	. . . . 5 |- ( xSingleton { x } <-> { x } = { x } )
8	4, 7	mpbir 214	. . . 4 |- xSingleton { x }
9		fnsingle 30734	. . . . 5 |- Singleton Fn _V
10		fnbrfvb 5927	. . . . 5 |- ( (Singleton Fn _V /\ x e. _V ) -> ( (Singleton ` x ) = { x } <-> xSingleton { x } ) )
11	9, 5, 10	mp2an 683	. . . 4 |- ( (Singleton ` x ) = { x } <-> xSingleton { x } )
12	8, 11	mpbir 214	. . 3 |- (Singleton ` x ) = { x }
13	3, 12	vtoclg 3118	. 2 |- ( A e. _V -> (Singleton ` A ) = { A } )
14		fvprc 5881	. . 3 |- ( -. A e. _V -> (Singleton ` A ) = (/) )
15		snprc 4047	. . . 4 |- ( -. A e. _V <-> { A } = (/) )
16	15	biimpi 199	. . 3 |- ( -. A e. _V -> { A } = (/) )
17	14, 16	eqtr4d 2498	. 2 |- ( -. A e. _V -> (Singleton ` A ) = { A } )
18	13, 17	pm2.61i 169	1 |- (Singleton ` A ) = { A }

Syntax hints:   -. wn 3   <-> wb 189   = wceq 1454   e. wcel 1897   _Vcvv 3056   (/)c0 3742   {csn 3979   class class class wbr 4415   Fn wfn 5595   ` cfv 5600  Singletoncsingle 30652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-symdif 3674  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-eprel 4763  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fo 5606  df-fv 5608  df-1st 6819  df-2nd 6820  df-txp 30668  df-singleton 30676

This theorem is referenced by: (None)