Theorem dfiota4 4372

Description: Alternate definition of iota in terms of 1_c. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
dfiota4	|- (iotaxph) = U.U.(1c i^i {{x | ph}})

Proof of Theorem dfiota4

Step	Hyp	Ref	Expression
1		iotauni 4351	. . 3 |- (E!xph -> (iotaxph) = U.{x | ph})
2		dfeu2 4333	. . . . . . . 8 |- (E!xph <-> {x | ph} e. 1c)
3		snssi 3852	. . . . . . . 8 |- ({x | ph} e. 1c -> {{x | ph}} C_ 1c)
4	2, 3	sylbi 187	. . . . . . 7 |- (E!xph -> {{x | ph}} C_ 1c)
5		df-ss 3259	. . . . . . . 8 |- ({{x | ph}} C_ 1c <-> ({{x | ph}} i^i 1c) = {{x | ph}})
6		incom 3448	. . . . . . . . 9 |- ({{x | ph}} i^i 1c) = (1c i^i {{x | ph}})
7	6	eqeq1i 2360	. . . . . . . 8 |- (({{x | ph}} i^i 1c) = {{x | ph}} <-> (1c i^i {{x | ph}}) = {{x | ph}})
8	5, 7	bitri 240	. . . . . . 7 |- ({{x | ph}} C_ 1c <-> (1c i^i {{x | ph}}) = {{x | ph}})
9	4, 8	sylib 188	. . . . . 6 |- (E!xph -> (1c i^i {{x | ph}}) = {{x | ph}})
10	9	unieqd 3902	. . . . 5 |- (E!xph -> U.(1c i^i {{x | ph}}) = U.{{x | ph}})
11		euabex 4334	. . . . . 6 |- (E!xph -> {x | ph} e. _V)
12		unisng 3908	. . . . . 6 |- ({x | ph} e. _V -> U.{{x | ph}} = {x | ph})
13	11, 12	syl 15	. . . . 5 |- (E!xph -> U.{{x | ph}} = {x | ph})
14	10, 13	eqtrd 2385	. . . 4 |- (E!xph -> U.(1c i^i {{x | ph}}) = {x | ph})
15	14	unieqd 3902	. . 3 |- (E!xph -> U.U.(1c i^i {{x | ph}}) = U.{x | ph})
16	1, 15	eqtr4d 2388	. 2 |- (E!xph -> (iotaxph) = U.U.(1c i^i {{x | ph}}))
17		iotanul 4354	. . 3 |- (-. E!xph -> (iotaxph) = (/))
18	2	notbii 287	. . . . . . . 8 |- (-. E!xph <-> -. {x | ph} e. 1c)
19		disjsn 3786	. . . . . . . 8 |- ((1c i^i {{x | ph}}) = (/) <-> -. {x | ph} e. 1c)
20	18, 19	bitr4i 243	. . . . . . 7 |- (-. E!xph <-> (1c i^i {{x | ph}}) = (/))
21	20	biimpi 186	. . . . . 6 |- (-. E!xph -> (1c i^i {{x | ph}}) = (/))
22	21	unieqd 3902	. . . . 5 |- (-. E!xph -> U.(1c i^i {{x | ph}}) = U.(/))
23	22	unieqd 3902	. . . 4 |- (-. E!xph -> U.U.(1c i^i {{x | ph}}) = U.U.(/))
24		uni0 3918	. . . . . 6 |- U.(/) = (/)
25	24	unieqi 3901	. . . . 5 |- U.U.(/) = U.(/)
26	25, 24	eqtri 2373	. . . 4 |- U.U.(/) = (/)
27	23, 26	syl6eq 2401	. . 3 |- (-. E!xph -> U.U.(1c i^i {{x | ph}}) = (/))
28	17, 27	eqtr4d 2388	. 2 |- (-. E!xph -> (iotaxph) = U.U.(1c i^i {{x | ph}}))
29	16, 28	pm2.61i 156	1 |- (iotaxph) = U.U.(1c i^i {{x | ph}})

Syntax hints:  -. wn 3   = wceq 1642   e. wcel 1710  E!weu 2204  {cab 2339  _Vcvv 2859   i^i cin 3208   C_ wss 3257  (/)c0 3550  {csn 3737  U.cuni 3891  1_cc1c 4134  iotacio 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-1c 4136  df-iota 4339

This theorem is referenced by: (None)