Theorem dfiota3 30475

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

Assertion

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

Proof of Theorem dfiota3

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

Step	Hyp	Ref	Expression
1		df-iota 5565	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		abeq1 2554	. . . . 5 |- ( { 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 1827	. . . . . 6 |- ( 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 3090	. . . . . . . . 9 |- y e. _V
5		sneq 4012	. . . . . . . . . 10 |- ( w = y -> { w } = { y } )
6	5	eqeq2d 2443	. . . . . . . . 9 |- ( w = y -> ( { x | ph } = { w } <-> { x | ph } = { y } ) )
7	4, 6	ceqsexv 3124	. . . . . . . 8 |- ( E. w ( w = y /\ { x | ph } = { w } ) <-> { x | ph } = { y } )
8		snex 4663	. . . . . . . . . . 11 |- { w } e. _V
9		eqeq1 2433	. . . . . . . . . . . . 13 |- ( z = { w } -> ( z = { x | ph } <-> { w } = { x | ph } ) )
10		eleq2 2502	. . . . . . . . . . . . 13 |- ( z = { w } -> ( y e. z <-> y e. { w } ) )
11	9, 10	anbi12d 715	. . . . . . . . . . . 12 |- ( z = { w } -> ( ( z = { x | ph } /\ y e. z ) <-> ( { w } = { x | ph } /\ y e. { w } ) ) )
12		eqcom 2438	. . . . . . . . . . . . 13 |- ( { w } = { x | ph } <-> { x | ph } = { w } )
13		elsn 4016	. . . . . . . . . . . . . 14 |- ( y e. { w } <-> y = w )
14		equcom 1846	. . . . . . . . . . . . . 14 |- ( y = w <-> w = y )
15	13, 14	bitri 252	. . . . . . . . . . . . 13 |- ( y e. { w } <-> w = y )
16	12, 15	anbi12ci 702	. . . . . . . . . . . 12 |- ( ( { w } = { x | ph } /\ y e. { w } ) <-> ( w = y /\ { x | ph } = { w } ) )
17	11, 16	syl6bb 264	. . . . . . . . . . 11 |- ( z = { w } -> ( ( z = { x | ph } /\ y e. z ) <-> ( w = y /\ { x | ph } = { w } ) ) )
18	8, 17	ceqsexv 3124	. . . . . . . . . 10 |- ( E. z ( z = { w } /\ ( z = { x | ph } /\ y e. z ) ) <-> ( w = y /\ { x | ph } = { w } ) )
19		an13 806	. . . . . . . . . . 11 |- ( ( z = { w } /\ ( z = { x | ph } /\ y e. z ) ) <-> ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
20	19	exbii 1714	. . . . . . . . . 10 |- ( E. z ( z = { w } /\ ( z = { x | ph } /\ y e. z ) ) <-> E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
21	18, 20	bitr3i 254	. . . . . . . . 9 |- ( ( w = y /\ { x | ph } = { w } ) <-> E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
22	21	exbii 1714	. . . . . . . 8 |- ( E. w ( w = y /\ { x | ph } = { w } ) <-> E. w E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
23	7, 22	bitr3i 254	. . . . . . 7 |- ( { x | ph } = { y } <-> E. w E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
24		excom 1901	. . . . . . 7 |- ( 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 252	. . . . . 6 |- ( { x | ph } = { y } <-> E. z E. w ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
26		eluniab 4233	. . . . . 6 |- ( 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 280	. . . . 5 |- ( { x | ph } = { y } <-> y e. U. { z | E. w ( z = { x | ph } /\ z = { w } ) } )
28	2, 27	mpgbir 1669	. . . 4 |- { y | { x | ph } = { y } } = U. { z | E. w ( z = { x | ph } /\ z = { w } ) }
29		df-sn 4003	. . . . . . 7 |- { { x | ph } } = { z | z = { x | ph } }
30		dfsingles2 30473	. . . . . . 7 |- Singletons = { z | E. w z = { w } }
31	29, 30	ineq12i 3668	. . . . . 6 |- ( { { x | ph } } i^i Singletons ) = ( { z | z = { x | ph } } i^i { z | E. w z = { w } } )
32		inab 3747	. . . . . . 7 |- ( { z | z = { x | ph } } i^i { z | E. w z = { w } } ) = { z | ( z = { x | ph } /\ E. w z = { w } ) }
33		19.42v 1826	. . . . . . . . 9 |- ( E. w ( z = { x | ph } /\ z = { w } ) <-> ( z = { x | ph } /\ E. w z = { w } ) )
34	33	bicomi 205	. . . . . . . 8 |- ( ( z = { x | ph } /\ E. w z = { w } ) <-> E. w ( z = { x | ph } /\ z = { w } ) )
35	34	abbii 2563	. . . . . . 7 |- { z | ( z = { x | ph } /\ E. w z = { w } ) } = { z | E. w ( z = { x | ph } /\ z = { w } ) }
36	32, 35	eqtri 2458	. . . . . 6 |- ( { z | z = { x | ph } } i^i { z | E. w z = { w } } ) = { z | E. w ( z = { x | ph } /\ z = { w } ) }
37	31, 36	eqtri 2458	. . . . 5 |- ( { { x | ph } } i^i Singletons ) = { z | E. w ( z = { x | ph } /\ z = { w } ) }
38	37	unieqi 4231	. . . 4 |- U. ( { { x | ph } } i^i Singletons ) = U. { z | E. w ( z = { x | ph } /\ z = { w } ) }
39	28, 38	eqtr4i 2461	. . 3 |- { y | { x | ph } = { y } } = U. ( { { x | ph } } i^i Singletons )
40	39	unieqi 4231	. 2 |- U. { y | { x | ph } = { y } } = U. U. ( { { x | ph } } i^i Singletons )
41	1, 40	eqtri 2458	1 |- ( iota x ph ) = U. U. ( { { x | ph } } i^i Singletons )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1659   e. wcel 1870   {cab 2414   i^i cin 3441   {csn 4002   U.cuni 4222   iotacio 5563   Singletonscsingles 30390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-symdif 3699  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-eprel 4765  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6807  df-2nd 6808  df-txp 30405  df-singleton 30413  df-singles 30414

This theorem is referenced by:  dffv5  30476