Theorem dfiota3 27954

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 5381	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		abeq1 2549	. . . . 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 1925	. . . . . 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 2975	. . . . . . . . 9 |- y e. _V
5		sneq 3887	. . . . . . . . . 10 |- ( w = y -> { w } = { y } )
6	5	eqeq2d 2454	. . . . . . . . 9 |- ( w = y -> ( { x | ph } = { w } <-> { x | ph } = { y } ) )
7	4, 6	ceqsexv 3009	. . . . . . . 8 |- ( E. w ( w = y /\ { x | ph } = { w } ) <-> { x | ph } = { y } )
8		snex 4533	. . . . . . . . . . 11 |- { w } e. _V
9		eqeq1 2449	. . . . . . . . . . . . 13 |- ( z = { w } -> ( z = { x | ph } <-> { w } = { x | ph } ) )
10		eleq2 2504	. . . . . . . . . . . . 13 |- ( z = { w } -> ( y e. z <-> y e. { w } ) )
11	9, 10	anbi12d 710	. . . . . . . . . . . 12 |- ( z = { w } -> ( ( z = { x | ph } /\ y e. z ) <-> ( { w } = { x | ph } /\ y e. { w } ) ) )
12		eqcom 2445	. . . . . . . . . . . . 13 |- ( { w } = { x | ph } <-> { x | ph } = { w } )
13		elsn 3891	. . . . . . . . . . . . . 14 |- ( y e. { w } <-> y = w )
14		equcom 1732	. . . . . . . . . . . . . 14 |- ( y = w <-> w = y )
15	13, 14	bitri 249	. . . . . . . . . . . . 13 |- ( y e. { w } <-> w = y )
16	12, 15	anbi12ci 698	. . . . . . . . . . . 12 |- ( ( { w } = { x | ph } /\ y e. { w } ) <-> ( w = y /\ { x | ph } = { w } ) )
17	11, 16	syl6bb 261	. . . . . . . . . . 11 |- ( z = { w } -> ( ( z = { x | ph } /\ y e. z ) <-> ( w = y /\ { x | ph } = { w } ) ) )
18	8, 17	ceqsexv 3009	. . . . . . . . . 10 |- ( E. z ( z = { w } /\ ( z = { x | ph } /\ y e. z ) ) <-> ( w = y /\ { x | ph } = { w } ) )
19		an13 797	. . . . . . . . . . 11 |- ( ( z = { w } /\ ( z = { x | ph } /\ y e. z ) ) <-> ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
20	19	exbii 1634	. . . . . . . . . 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 251	. . . . . . . . 9 |- ( ( w = y /\ { x | ph } = { w } ) <-> E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
22	21	exbii 1634	. . . . . . . 8 |- ( E. w ( w = y /\ { x | ph } = { w } ) <-> E. w E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
23	7, 22	bitr3i 251	. . . . . . 7 |- ( { x | ph } = { y } <-> E. w E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
24		excom 1787	. . . . . . 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 249	. . . . . 6 |- ( { x | ph } = { y } <-> E. z E. w ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
26		eluniab 4102	. . . . . 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 277	. . . . 5 |- ( { x | ph } = { y } <-> y e. U. { z | E. w ( z = { x | ph } /\ z = { w } ) } )
28	2, 27	mpgbir 1595	. . . 4 |- { y | { x | ph } = { y } } = U. { z | E. w ( z = { x | ph } /\ z = { w } ) }
29		df-sn 3878	. . . . . . 7 |- { { x | ph } } = { z | z = { x | ph } }
30		dfsingles2 27952	. . . . . . 7 |- Singletons = { z | E. w z = { w } }
31	29, 30	ineq12i 3550	. . . . . 6 |- ( { { x | ph } } i^i Singletons ) = ( { z | z = { x | ph } } i^i { z | E. w z = { w } } )
32		inab 3618	. . . . . . 7 |- ( { z | z = { x | ph } } i^i { z | E. w z = { w } } ) = { z | ( z = { x | ph } /\ E. w z = { w } ) }
33		19.42v 1924	. . . . . . . . 9 |- ( E. w ( z = { x | ph } /\ z = { w } ) <-> ( z = { x | ph } /\ E. w z = { w } ) )
34	33	bicomi 202	. . . . . . . 8 |- ( ( z = { x | ph } /\ E. w z = { w } ) <-> E. w ( z = { x | ph } /\ z = { w } ) )
35	34	abbii 2555	. . . . . . 7 |- { z | ( z = { x | ph } /\ E. w z = { w } ) } = { z | E. w ( z = { x | ph } /\ z = { w } ) }
36	32, 35	eqtri 2463	. . . . . 6 |- ( { z | z = { x | ph } } i^i { z | E. w z = { w } } ) = { z | E. w ( z = { x | ph } /\ z = { w } ) }
37	31, 36	eqtri 2463	. . . . 5 |- ( { { x | ph } } i^i Singletons ) = { z | E. w ( z = { x | ph } /\ z = { w } ) }
38	37	unieqi 4100	. . . 4 |- U. ( { { x | ph } } i^i Singletons ) = U. { z | E. w ( z = { x | ph } /\ z = { w } ) }
39	28, 38	eqtr4i 2466	. . 3 |- { y | { x | ph } = { y } } = U. ( { { x | ph } } i^i Singletons )
40	39	unieqi 4100	. 2 |- U. { y | { x | ph } = { y } } = U. U. ( { { x | ph } } i^i Singletons )
41	1, 40	eqtri 2463	1 |- ( iota x ph ) = U. U. ( { { x | ph } } i^i Singletons )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   {cab 2429   i^i cin 3327   {csn 3877   U.cuni 4091   iotacio 5379   Singletonscsingles 27869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-eprel 4632  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-1st 6577  df-2nd 6578  df-symdif 27849  df-txp 27884  df-singleton 27892  df-singles 27893

This theorem is referenced by:  dffv5  27955