Theorem dfiota3 30761

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 5553	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		abeq1 2581	. . . . 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 1843	. . . . . 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 3034	. . . . . . . . 9 |- y e. _V
5		sneq 3969	. . . . . . . . . 10 |- ( w = y -> { w } = { y } )
6	5	eqeq2d 2481	. . . . . . . . 9 |- ( w = y -> ( { x | ph } = { w } <-> { x | ph } = { y } ) )
7	4, 6	ceqsexv 3070	. . . . . . . 8 |- ( E. w ( w = y /\ { x | ph } = { w } ) <-> { x | ph } = { y } )
8		snex 4641	. . . . . . . . . . 11 |- { w } e. _V
9		eqeq1 2475	. . . . . . . . . . . . 13 |- ( z = { w } -> ( z = { x | ph } <-> { w } = { x | ph } ) )
10		eleq2 2538	. . . . . . . . . . . . 13 |- ( z = { w } -> ( y e. z <-> y e. { w } ) )
11	9, 10	anbi12d 725	. . . . . . . . . . . 12 |- ( z = { w } -> ( ( z = { x | ph } /\ y e. z ) <-> ( { w } = { x | ph } /\ y e. { w } ) ) )
12		eqcom 2478	. . . . . . . . . . . . 13 |- ( { w } = { x | ph } <-> { x | ph } = { w } )
13		elsn 3973	. . . . . . . . . . . . . 14 |- ( y e. { w } <-> y = w )
14		equcom 1870	. . . . . . . . . . . . . 14 |- ( y = w <-> w = y )
15	13, 14	bitri 257	. . . . . . . . . . . . 13 |- ( y e. { w } <-> w = y )
16	12, 15	anbi12ci 712	. . . . . . . . . . . 12 |- ( ( { w } = { x | ph } /\ y e. { w } ) <-> ( w = y /\ { x | ph } = { w } ) )
17	11, 16	syl6bb 269	. . . . . . . . . . 11 |- ( z = { w } -> ( ( z = { x | ph } /\ y e. z ) <-> ( w = y /\ { x | ph } = { w } ) ) )
18	8, 17	ceqsexv 3070	. . . . . . . . . 10 |- ( E. z ( z = { w } /\ ( z = { x | ph } /\ y e. z ) ) <-> ( w = y /\ { x | ph } = { w } ) )
19		an13 816	. . . . . . . . . . 11 |- ( ( z = { w } /\ ( z = { x | ph } /\ y e. z ) ) <-> ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
20	19	exbii 1726	. . . . . . . . . 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 259	. . . . . . . . 9 |- ( ( w = y /\ { x | ph } = { w } ) <-> E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
22	21	exbii 1726	. . . . . . . 8 |- ( E. w ( w = y /\ { x | ph } = { w } ) <-> E. w E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
23	7, 22	bitr3i 259	. . . . . . 7 |- ( { x | ph } = { y } <-> E. w E. z ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
24		excom 1944	. . . . . . 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 257	. . . . . 6 |- ( { x | ph } = { y } <-> E. z E. w ( y e. z /\ ( z = { x | ph } /\ z = { w } ) ) )
26		eluniab 4201	. . . . . 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 285	. . . . 5 |- ( { x | ph } = { y } <-> y e. U. { z | E. w ( z = { x | ph } /\ z = { w } ) } )
28	2, 27	mpgbir 1681	. . . 4 |- { y | { x | ph } = { y } } = U. { z | E. w ( z = { x | ph } /\ z = { w } ) }
29		df-sn 3960	. . . . . . 7 |- { { x | ph } } = { z | z = { x | ph } }
30		dfsingles2 30759	. . . . . . 7 |- Singletons = { z | E. w z = { w } }
31	29, 30	ineq12i 3623	. . . . . 6 |- ( { { x | ph } } i^i Singletons ) = ( { z | z = { x | ph } } i^i { z | E. w z = { w } } )
32		inab 3702	. . . . . . 7 |- ( { z | z = { x | ph } } i^i { z | E. w z = { w } } ) = { z | ( z = { x | ph } /\ E. w z = { w } ) }
33		19.42v 1842	. . . . . . . . 9 |- ( E. w ( z = { x | ph } /\ z = { w } ) <-> ( z = { x | ph } /\ E. w z = { w } ) )
34	33	bicomi 207	. . . . . . . 8 |- ( ( z = { x | ph } /\ E. w z = { w } ) <-> E. w ( z = { x | ph } /\ z = { w } ) )
35	34	abbii 2587	. . . . . . 7 |- { z | ( z = { x | ph } /\ E. w z = { w } ) } = { z | E. w ( z = { x | ph } /\ z = { w } ) }
36	32, 35	eqtri 2493	. . . . . 6 |- ( { z | z = { x | ph } } i^i { z | E. w z = { w } } ) = { z | E. w ( z = { x | ph } /\ z = { w } ) }
37	31, 36	eqtri 2493	. . . . 5 |- ( { { x | ph } } i^i Singletons ) = { z | E. w ( z = { x | ph } /\ z = { w } ) }
38	37	unieqi 4199	. . . 4 |- U. ( { { x | ph } } i^i Singletons ) = U. { z | E. w ( z = { x | ph } /\ z = { w } ) }
39	28, 38	eqtr4i 2496	. . 3 |- { y | { x | ph } = { y } } = U. ( { { x | ph } } i^i Singletons )
40	39	unieqi 4199	. 2 |- U. { y | { x | ph } = { y } } = U. U. ( { { x | ph } } i^i Singletons )
41	1, 40	eqtri 2493	1 |- ( iota x ph ) = U. U. ( { { x | ph } } i^i Singletons )

Syntax hints:   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   {cab 2457   i^i cin 3389   {csn 3959   U.cuni 4190   iotacio 5551   Singletonscsingles 30676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-eprel 4750  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812  df-2nd 6813  df-txp 30691  df-singleton 30699  df-singles 30700

This theorem is referenced by:  dffv5  30762