Theorem fnsingle 32026

Description: The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fnsingle	|- Singleton Fn _V

Proof of Theorem fnsingle

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 3737	. . . . 5 |- ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( _I (x) _V ) ) ) C_ ( _V X. _V )
2		df-rel 5121	. . . . 5 |- ( Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( _I (x) _V ) ) ) <-> ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( _I (x) _V ) ) ) C_ ( _V X. _V ) )
3	1, 2	mpbir 221	. . . 4 |- Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( _I (x) _V ) ) )
4		df-singleton 31969	. . . . 5 |- Singleton = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( _I (x) _V ) ) )
5	4	releqi 5202	. . . 4 |- ( Rel Singleton <-> Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( _I (x) _V ) ) ) )
6	3, 5	mpbir 221	. . 3 |- Rel Singleton
7		vex 3203	. . . . . . 7 |- x e. _V
8		vex 3203	. . . . . . 7 |- y e. _V
9	7, 8	brsingle 32024	. . . . . 6 |- ( xSingleton y <-> y = { x } )
10		vex 3203	. . . . . . 7 |- z e. _V
11	7, 10	brsingle 32024	. . . . . 6 |- ( xSingleton z <-> z = { x } )
12		eqtr3 2643	. . . . . 6 |- ( ( y = { x } /\ z = { x } ) -> y = z )
13	9, 11, 12	syl2anb 496	. . . . 5 |- ( ( xSingleton y /\ xSingleton z ) -> y = z )
14	13	ax-gen 1722	. . . 4 |- A. z ( ( xSingleton y /\ xSingleton z ) -> y = z )
15	14	gen2 1723	. . 3 |- A. x A. y A. z ( ( xSingleton y /\ xSingleton z ) -> y = z )
16		dffun2 5898	. . 3 |- ( Fun Singleton <-> ( Rel Singleton /\ A. x A. y A. z ( ( xSingleton y /\ xSingleton z ) -> y = z ) ) )
17	6, 15, 16	mpbir2an 955	. 2 |- Fun Singleton
18		eqv 3205	. . 3 |- ( dom Singleton = _V <-> A. x x e. dom Singleton )
19		eqid 2622	. . . . . 6 |- { x } = { x }
20		snex 4908	. . . . . . 7 |- { x } e. _V
21	7, 20	brsingle 32024	. . . . . 6 |- ( xSingleton { x } <-> { x } = { x } )
22	19, 21	mpbir 221	. . . . 5 |- xSingleton { x }
23		breq2 4657	. . . . . 6 |- ( y = { x } -> ( xSingleton y <-> xSingleton { x } ) )
24	20, 23	spcev 3300	. . . . 5 |- ( xSingleton { x } -> E. y xSingleton y )
25	22, 24	ax-mp 5	. . . 4 |- E. y xSingleton y
26	7	eldm 5321	. . . 4 |- ( x e. dom Singleton <-> E. y xSingleton y )
27	25, 26	mpbir 221	. . 3 |- x e. dom Singleton
28	18, 27	mpgbir 1726	. 2 |- dom Singleton = _V
29		df-fn 5891	. 2 |- (Singleton Fn _V <-> ( Fun Singleton /\ dom Singleton = _V ) )
30	17, 28, 29	mpbir2an 955	1 |- Singleton Fn _V

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   /_\ csymdif 3843   {csn 4177   class class class wbr 4653   _I cid 5023   _E cep 5028   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   (x) ctxp 31937  Singletoncsingle 31945

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  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-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969

This theorem is referenced by:  fvsingle  32027