Theorem fvsingle 29801

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 5848	. . . 4 |- ( x = A -> (Singleton ` x ) = (Singleton ` A ) )
2		sneq 4026	. . . 4 |- ( x = A -> { x } = { A } )
3	1, 2	eqeq12d 2476	. . 3 |- ( x = A -> ( (Singleton ` x ) = { x } <-> (Singleton ` A ) = { A } ) )
4		eqid 2454	. . . . 5 |- { x } = { x }
5		vex 3109	. . . . . 6 |- x e. _V
6		snex 4678	. . . . . 6 |- { x } e. _V
7	5, 6	brsingle 29798	. . . . 5 |- ( xSingleton { x } <-> { x } = { x } )
8	4, 7	mpbir 209	. . . 4 |- xSingleton { x }
9		fnsingle 29800	. . . . 5 |- Singleton Fn _V
10		fnbrfvb 5888	. . . . 5 |- ( (Singleton Fn _V /\ x e. _V ) -> ( (Singleton ` x ) = { x } <-> xSingleton { x } ) )
11	9, 5, 10	mp2an 670	. . . 4 |- ( (Singleton ` x ) = { x } <-> xSingleton { x } )
12	8, 11	mpbir 209	. . 3 |- (Singleton ` x ) = { x }
13	3, 12	vtoclg 3164	. 2 |- ( A e. _V -> (Singleton ` A ) = { A } )
14		fvprc 5842	. . 3 |- ( -. A e. _V -> (Singleton ` A ) = (/) )
15		snprc 4079	. . . 4 |- ( -. A e. _V <-> { A } = (/) )
16	15	biimpi 194	. . 3 |- ( -. A e. _V -> { A } = (/) )
17	14, 16	eqtr4d 2498	. 2 |- ( -. A e. _V -> (Singleton ` A ) = { A } )
18	13, 17	pm2.61i 164	1 |- (Singleton ` A ) = { A }

Syntax hints:   -. wn 3   <-> wb 184   = wceq 1398   e. wcel 1823   _Vcvv 3106   (/)c0 3783   {csn 4016   class class class wbr 4439   Fn wfn 5565   ` cfv 5570  Singletoncsingle 29718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-symdif 3715  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-eprel 4780  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773  df-2nd 6774  df-txp 29734  df-singleton 29742

This theorem is referenced by: (None)