Theorem dfiota4 4373

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

Assertion

Ref	Expression
dfiota4	⊢ (℩xφ) = ∪∪(1c ∩ {{x ∣ φ}})

Proof of Theorem dfiota4

Step	Hyp	Ref	Expression
1		iotauni 4352	. . 3 ⊢ (∃!xφ → (℩xφ) = ∪{x ∣ φ})
2		dfeu2 4334	. . . . . . . 8 ⊢ (∃!xφ ↔ {x ∣ φ} ∈ 1c)
3		snssi 3853	. . . . . . . 8 ⊢ ({x ∣ φ} ∈ 1c → {{x ∣ φ}} ⊆ 1c)
4	2, 3	sylbi 187	. . . . . . 7 ⊢ (∃!xφ → {{x ∣ φ}} ⊆ 1c)
5		df-ss 3260	. . . . . . . 8 ⊢ ({{x ∣ φ}} ⊆ 1c ↔ ({{x ∣ φ}} ∩ 1c) = {{x ∣ φ}})
6		incom 3449	. . . . . . . . 9 ⊢ ({{x ∣ φ}} ∩ 1c) = (1c ∩ {{x ∣ φ}})
7	6	eqeq1i 2360	. . . . . . . 8 ⊢ (({{x ∣ φ}} ∩ 1c) = {{x ∣ φ}} ↔ (1c ∩ {{x ∣ φ}}) = {{x ∣ φ}})
8	5, 7	bitri 240	. . . . . . 7 ⊢ ({{x ∣ φ}} ⊆ 1c ↔ (1c ∩ {{x ∣ φ}}) = {{x ∣ φ}})
9	4, 8	sylib 188	. . . . . 6 ⊢ (∃!xφ → (1c ∩ {{x ∣ φ}}) = {{x ∣ φ}})
10	9	unieqd 3903	. . . . 5 ⊢ (∃!xφ → ∪(1c ∩ {{x ∣ φ}}) = ∪{{x ∣ φ}})
11		euabex 4335	. . . . . 6 ⊢ (∃!xφ → {x ∣ φ} ∈ V)
12		unisng 3909	. . . . . 6 ⊢ ({x ∣ φ} ∈ V → ∪{{x ∣ φ}} = {x ∣ φ})
13	11, 12	syl 15	. . . . 5 ⊢ (∃!xφ → ∪{{x ∣ φ}} = {x ∣ φ})
14	10, 13	eqtrd 2385	. . . 4 ⊢ (∃!xφ → ∪(1c ∩ {{x ∣ φ}}) = {x ∣ φ})
15	14	unieqd 3903	. . 3 ⊢ (∃!xφ → ∪∪(1c ∩ {{x ∣ φ}}) = ∪{x ∣ φ})
16	1, 15	eqtr4d 2388	. 2 ⊢ (∃!xφ → (℩xφ) = ∪∪(1c ∩ {{x ∣ φ}}))
17		iotanul 4355	. . 3 ⊢ (¬ ∃!xφ → (℩xφ) = ∅)
18	2	notbii 287	. . . . . . . 8 ⊢ (¬ ∃!xφ ↔ ¬ {x ∣ φ} ∈ 1c)
19		disjsn 3787	. . . . . . . 8 ⊢ ((1c ∩ {{x ∣ φ}}) = ∅ ↔ ¬ {x ∣ φ} ∈ 1c)
20	18, 19	bitr4i 243	. . . . . . 7 ⊢ (¬ ∃!xφ ↔ (1c ∩ {{x ∣ φ}}) = ∅)
21	20	biimpi 186	. . . . . 6 ⊢ (¬ ∃!xφ → (1c ∩ {{x ∣ φ}}) = ∅)
22	21	unieqd 3903	. . . . 5 ⊢ (¬ ∃!xφ → ∪(1c ∩ {{x ∣ φ}}) = ∪∅)
23	22	unieqd 3903	. . . 4 ⊢ (¬ ∃!xφ → ∪∪(1c ∩ {{x ∣ φ}}) = ∪∪∅)
24		uni0 3919	. . . . . 6 ⊢ ∪∅ = ∅
25	24	unieqi 3902	. . . . 5 ⊢ ∪∪∅ = ∪∅
26	25, 24	eqtri 2373	. . . 4 ⊢ ∪∪∅ = ∅
27	23, 26	syl6eq 2401	. . 3 ⊢ (¬ ∃!xφ → ∪∪(1c ∩ {{x ∣ φ}}) = ∅)
28	17, 27	eqtr4d 2388	. 2 ⊢ (¬ ∃!xφ → (℩xφ) = ∪∪(1c ∩ {{x ∣ φ}}))
29	16, 28	pm2.61i 156	1 ⊢ (℩xφ) = ∪∪(1c ∩ {{x ∣ φ}})

Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551  {csn 3738  ∪cuni 3892  1_cc1c 4135  ℩cio 4338

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-uni 3893  df-1c 4137  df-iota 4340

This theorem is referenced by: (None)