Theorem dfiota3 30632

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 5501	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		abeq1 2532	. . . . 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 1828	. . . . . 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 3019	. . . . . . . . 9 |- y e. _V
5		sneq 3944	. . . . . . . . . 10 |- ( w = y -> { w } = { y } )
6	5	eqeq2d 2432	. . . . . . . . 9 |- ( w = y -> ( { x | ph } = { w } <-> { x | ph } = { y } ) )
7	4, 6	ceqsexv 3054	. . . . . . . 8 |- ( E. w ( w = y /\ { x | ph } = { w } ) <-> { x | ph } = { y } )
8		snex 4598	. . . . . . . . . . 11 |- { w } e. _V
9		eqeq1 2426	. . . . . . . . . . . . 13 |- ( z = { w } -> ( z = { x | ph } <-> { w } = { x | ph } ) )
10		eleq2 2489	. . . . . . . . . . . . 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 2429	. . . . . . . . . . . . 13 |- ( { w } = { x | ph } <-> { x | ph } = { w } )
13		elsn 3948	. . . . . . . . . . . . . 14 |- ( y e. { w } <-> y = w )
14		equcom 1848	. . . . . . . . . . . . . 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 3054	. . . . . . . . . 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 1712	. . . . . . . . . 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 1712	. . . . . . . 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 1903	. . . . . . 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 4166	. . . . . 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 1667	. . . 4 |- { y | { x | ph } = { y } } = U. { z | E. w ( z = { x | ph } /\ z = { w } ) }
29		df-sn 3935	. . . . . . 7 |- { { x | ph } } = { z | z = { x | ph } }
30		dfsingles2 30630	. . . . . . 7 |- Singletons = { z | E. w z = { w } }
31	29, 30	ineq12i 3598	. . . . . 6 |- ( { { x | ph } } i^i Singletons ) = ( { z | z = { x | ph } } i^i { z | E. w z = { w } } )
32		inab 3677	. . . . . . 7 |- ( { z | z = { x | ph } } i^i { z | E. w z = { w } } ) = { z | ( z = { x | ph } /\ E. w z = { w } ) }
33		19.42v 1827	. . . . . . . . 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 2538	. . . . . . 7 |- { z | ( z = { x | ph } /\ E. w z = { w } ) } = { z | E. w ( z = { x | ph } /\ z = { w } ) }
36	32, 35	eqtri 2444	. . . . . 6 |- ( { z | z = { x | ph } } i^i { z | E. w z = { w } } ) = { z | E. w ( z = { x | ph } /\ z = { w } ) }
37	31, 36	eqtri 2444	. . . . 5 |- ( { { x | ph } } i^i Singletons ) = { z | E. w ( z = { x | ph } /\ z = { w } ) }
38	37	unieqi 4164	. . . 4 |- U. ( { { x | ph } } i^i Singletons ) = U. { z | E. w ( z = { x | ph } /\ z = { w } ) }
39	28, 38	eqtr4i 2447	. . 3 |- { y | { x | ph } = { y } } = U. ( { { x | ph } } i^i Singletons )
40	39	unieqi 4164	. 2 |- U. { y | { x | ph } = { y } } = U. U. ( { { x | ph } } i^i Singletons )
41	1, 40	eqtri 2444	1 |- ( iota x ph ) = U. U. ( { { x | ph } } i^i Singletons )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1657   e. wcel 1872   {cab 2408   i^i cin 3371   {csn 3934   U.cuni 4155   iotacio 5499   Singletonscsingles 30547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-sbc 3236  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-symdif 3629  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-opab 4419  df-mpt 4420  df-eprel 4700  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-fo 5543  df-fv 5545  df-1st 6744  df-2nd 6745  df-txp 30562  df-singleton 30570  df-singles 30571

This theorem is referenced by:  dffv5  30633