dfiota3

Description: A definition of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013)

		Ref	Expression
	Assertion	dfiota3	\|- ( iota x ph ) = U. U. ( { { x \| ph } } i^i Singletons )

Proof

Step	Hyp	Ref	Expression
1		df-iota	\|- ( iota x ph ) = U. { y \| { x \| ph } = { y } }
2		eqabcb	\|- ( { y \| { x \| ph } = { y } } = U. { z \| E. w ( z = { x \| ph } /\ z = { w } ) } <-> A. y ( { x \| ph } = { y } <-> y e. U. { z \| E. w ( z = { x \| ph } /\ z = { w } ) } ) )
3		exdistr	\|- ( E. z E. w ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) <-> E. z ( y e. z /\ E. w ( z = { x \| ph } /\ z = { w } ) ) )
4		vex	\|- y e. _V
5		sneq	\|- ( w = y -> { w } = { y } )
6	5	eqeq2d	\|- ( w = y -> ( { x \| ph } = { w } <-> { x \| ph } = { y } ) )
7	4 6	ceqsexv	\|- ( E. w ( w = y /\ { x \| ph } = { w } ) <-> { x \| ph } = { y } )
8		vsnex	\|- { w } e. _V
9		eqeq1	\|- ( z = { w } -> ( z = { x \| ph } <-> { w } = { x \| ph } ) )
10		eleq2	\|- ( z = { w } -> ( y e. z <-> y e. { w } ) )
11	9 10	anbi12d	\|- ( z = { w } -> ( ( z = { x \| ph } /\ y e. z ) <-> ( { w } = { x \| ph } /\ y e. { w } ) ) )
12		eqcom	\|- ( { w } = { x \| ph } <-> { x \| ph } = { w } )
13		velsn	\|- ( y e. { w } <-> y = w )
14		equcom	\|- ( y = w <-> w = y )
15	13 14	bitri	\|- ( y e. { w } <-> w = y )
16	12 15	anbi12ci	\|- ( ( { w } = { x \| ph } /\ y e. { w } ) <-> ( w = y /\ { x \| ph } = { w } ) )
17	11 16	bitrdi	\|- ( z = { w } -> ( ( z = { x \| ph } /\ y e. z ) <-> ( w = y /\ { x \| ph } = { w } ) ) )
18	8 17	ceqsexv	\|- ( E. z ( z = { w } /\ ( z = { x \| ph } /\ y e. z ) ) <-> ( w = y /\ { x \| ph } = { w } ) )
19		an13	\|- ( ( z = { w } /\ ( z = { x \| ph } /\ y e. z ) ) <-> ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) )
20	19	exbii	\|- ( E. z ( z = { w } /\ ( z = { x \| ph } /\ y e. z ) ) <-> E. z ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) )
21	18 20	bitr3i	\|- ( ( w = y /\ { x \| ph } = { w } ) <-> E. z ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) )
22	21	exbii	\|- ( E. w ( w = y /\ { x \| ph } = { w } ) <-> E. w E. z ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) )
23	7 22	bitr3i	\|- ( { x \| ph } = { y } <-> E. w E. z ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) )
24		excom	\|- ( E. w E. z ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) <-> E. z E. w ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) )
25	23 24	bitri	\|- ( { x \| ph } = { y } <-> E. z E. w ( y e. z /\ ( z = { x \| ph } /\ z = { w } ) ) )
26		eluniab	\|- ( y e. U. { z \| E. w ( z = { x \| ph } /\ z = { w } ) } <-> E. z ( y e. z /\ E. w ( z = { x \| ph } /\ z = { w } ) ) )
27	3 25 26	3bitr4i	\|- ( { x \| ph } = { y } <-> y e. U. { z \| E. w ( z = { x \| ph } /\ z = { w } ) } )
28	2 27	mpgbir	\|- { y \| { x \| ph } = { y } } = U. { z \| E. w ( z = { x \| ph } /\ z = { w } ) }
29		df-sn	\|- { { x \| ph } } = { z \| z = { x \| ph } }
30		dfsingles2	\|- Singletons = { z \| E. w z = { w } }
31	29 30	ineq12i	\|- ( { { x \| ph } } i^i Singletons ) = ( { z \| z = { x \| ph } } i^i { z \| E. w z = { w } } )
32		inab	\|- ( { z \| z = { x \| ph } } i^i { z \| E. w z = { w } } ) = { z \| ( z = { x \| ph } /\ E. w z = { w } ) }
33		19.42v	\|- ( E. w ( z = { x \| ph } /\ z = { w } ) <-> ( z = { x \| ph } /\ E. w z = { w } ) )
34	33	bicomi	\|- ( ( z = { x \| ph } /\ E. w z = { w } ) <-> E. w ( z = { x \| ph } /\ z = { w } ) )
35	34	abbii	\|- { z \| ( z = { x \| ph } /\ E. w z = { w } ) } = { z \| E. w ( z = { x \| ph } /\ z = { w } ) }
36	32 35	eqtri	\|- ( { z \| z = { x \| ph } } i^i { z \| E. w z = { w } } ) = { z \| E. w ( z = { x \| ph } /\ z = { w } ) }
37	31 36	eqtri	\|- ( { { x \| ph } } i^i Singletons ) = { z \| E. w ( z = { x \| ph } /\ z = { w } ) }
38	37	unieqi	\|- U. ( { { x \| ph } } i^i Singletons ) = U. { z \| E. w ( z = { x \| ph } /\ z = { w } ) }
39	28 38	eqtr4i	\|- { y \| { x \| ph } = { y } } = U. ( { { x \| ph } } i^i Singletons )
40	39	unieqi	\|- U. { y \| { x \| ph } = { y } } = U. U. ( { { x \| ph } } i^i Singletons )
41	1 40	eqtri	\|- ( iota x ph ) = U. U. ( { { x \| ph } } i^i Singletons )

Metamath Proof Explorer

Theorem dfiota3

Proof