Theorem dfiin2g 4000

Description: Alternate definition of indexed intersection when B is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)

Assertion

Ref	Expression
dfiin2g	⊢ (∀x ∈ A B ∈ C → ∩x ∈ A B = ∩{y ∣ ∃x ∈ A y = B})

Distinct variable groups:   y,A   y,B   x,y

Allowed substitution hints:   A(x)   B(x)   C(x,y)

Proof of Theorem dfiin2g

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 2619	. . . 4 ⊢ (∀x ∈ A w ∈ B ↔ ∀x(x ∈ A → w ∈ B))
2		df-ral 2619	. . . . . 6 ⊢ (∀x ∈ A B ∈ C ↔ ∀x(x ∈ A → B ∈ C))
3		eleq2 2414	. . . . . . . . . . . . 13 ⊢ (z = B → (w ∈ z ↔ w ∈ B))
4	3	biimprcd 216	. . . . . . . . . . . 12 ⊢ (w ∈ B → (z = B → w ∈ z))
5	4	alrimiv 1631	. . . . . . . . . . 11 ⊢ (w ∈ B → ∀z(z = B → w ∈ z))
6		eqid 2353	. . . . . . . . . . . 12 ⊢ B = B
7		eqeq1 2359	. . . . . . . . . . . . . 14 ⊢ (z = B → (z = B ↔ B = B))
8	7, 3	imbi12d 311	. . . . . . . . . . . . 13 ⊢ (z = B → ((z = B → w ∈ z) ↔ (B = B → w ∈ B)))
9	8	spcgv 2939	. . . . . . . . . . . 12 ⊢ (B ∈ C → (∀z(z = B → w ∈ z) → (B = B → w ∈ B)))
10	6, 9	mpii 39	. . . . . . . . . . 11 ⊢ (B ∈ C → (∀z(z = B → w ∈ z) → w ∈ B))
11	5, 10	impbid2 195	. . . . . . . . . 10 ⊢ (B ∈ C → (w ∈ B ↔ ∀z(z = B → w ∈ z)))
12	11	imim2i 13	. . . . . . . . 9 ⊢ ((x ∈ A → B ∈ C) → (x ∈ A → (w ∈ B ↔ ∀z(z = B → w ∈ z))))
13	12	pm5.74d 238	. . . . . . . 8 ⊢ ((x ∈ A → B ∈ C) → ((x ∈ A → w ∈ B) ↔ (x ∈ A → ∀z(z = B → w ∈ z))))
14	13	alimi 1559	. . . . . . 7 ⊢ (∀x(x ∈ A → B ∈ C) → ∀x((x ∈ A → w ∈ B) ↔ (x ∈ A → ∀z(z = B → w ∈ z))))
15		albi 1564	. . . . . . 7 ⊢ (∀x((x ∈ A → w ∈ B) ↔ (x ∈ A → ∀z(z = B → w ∈ z))) → (∀x(x ∈ A → w ∈ B) ↔ ∀x(x ∈ A → ∀z(z = B → w ∈ z))))
16	14, 15	syl 15	. . . . . 6 ⊢ (∀x(x ∈ A → B ∈ C) → (∀x(x ∈ A → w ∈ B) ↔ ∀x(x ∈ A → ∀z(z = B → w ∈ z))))
17	2, 16	sylbi 187	. . . . 5 ⊢ (∀x ∈ A B ∈ C → (∀x(x ∈ A → w ∈ B) ↔ ∀x(x ∈ A → ∀z(z = B → w ∈ z))))
18		df-ral 2619	. . . . . . . 8 ⊢ (∀x ∈ A (z = B → w ∈ z) ↔ ∀x(x ∈ A → (z = B → w ∈ z)))
19	18	albii 1566	. . . . . . 7 ⊢ (∀z∀x ∈ A (z = B → w ∈ z) ↔ ∀z∀x(x ∈ A → (z = B → w ∈ z)))
20		alcom 1737	. . . . . . 7 ⊢ (∀x∀z(x ∈ A → (z = B → w ∈ z)) ↔ ∀z∀x(x ∈ A → (z = B → w ∈ z)))
21	19, 20	bitr4i 243	. . . . . 6 ⊢ (∀z∀x ∈ A (z = B → w ∈ z) ↔ ∀x∀z(x ∈ A → (z = B → w ∈ z)))
22		r19.23v 2730	. . . . . . . 8 ⊢ (∀x ∈ A (z = B → w ∈ z) ↔ (∃x ∈ A z = B → w ∈ z))
23		vex 2862	. . . . . . . . . 10 ⊢ z ∈ V
24		eqeq1 2359	. . . . . . . . . . 11 ⊢ (y = z → (y = B ↔ z = B))
25	24	rexbidv 2635	. . . . . . . . . 10 ⊢ (y = z → (∃x ∈ A y = B ↔ ∃x ∈ A z = B))
26	23, 25	elab 2985	. . . . . . . . 9 ⊢ (z ∈ {y ∣ ∃x ∈ A y = B} ↔ ∃x ∈ A z = B)
27	26	imbi1i 315	. . . . . . . 8 ⊢ ((z ∈ {y ∣ ∃x ∈ A y = B} → w ∈ z) ↔ (∃x ∈ A z = B → w ∈ z))
28	22, 27	bitr4i 243	. . . . . . 7 ⊢ (∀x ∈ A (z = B → w ∈ z) ↔ (z ∈ {y ∣ ∃x ∈ A y = B} → w ∈ z))
29	28	albii 1566	. . . . . 6 ⊢ (∀z∀x ∈ A (z = B → w ∈ z) ↔ ∀z(z ∈ {y ∣ ∃x ∈ A y = B} → w ∈ z))
30		19.21v 1890	. . . . . . 7 ⊢ (∀z(x ∈ A → (z = B → w ∈ z)) ↔ (x ∈ A → ∀z(z = B → w ∈ z)))
31	30	albii 1566	. . . . . 6 ⊢ (∀x∀z(x ∈ A → (z = B → w ∈ z)) ↔ ∀x(x ∈ A → ∀z(z = B → w ∈ z)))
32	21, 29, 31	3bitr3ri 267	. . . . 5 ⊢ (∀x(x ∈ A → ∀z(z = B → w ∈ z)) ↔ ∀z(z ∈ {y ∣ ∃x ∈ A y = B} → w ∈ z))
33	17, 32	syl6bb 252	. . . 4 ⊢ (∀x ∈ A B ∈ C → (∀x(x ∈ A → w ∈ B) ↔ ∀z(z ∈ {y ∣ ∃x ∈ A y = B} → w ∈ z)))
34	1, 33	syl5bb 248	. . 3 ⊢ (∀x ∈ A B ∈ C → (∀x ∈ A w ∈ B ↔ ∀z(z ∈ {y ∣ ∃x ∈ A y = B} → w ∈ z)))
35	34	abbidv 2467	. 2 ⊢ (∀x ∈ A B ∈ C → {w ∣ ∀x ∈ A w ∈ B} = {w ∣ ∀z(z ∈ {y ∣ ∃x ∈ A y = B} → w ∈ z)})
36		df-iin 3972	. 2 ⊢ ∩x ∈ A B = {w ∣ ∀x ∈ A w ∈ B}
37		df-int 3927	. 2 ⊢ ∩{y ∣ ∃x ∈ A y = B} = {w ∣ ∀z(z ∈ {y ∣ ∃x ∈ A y = B} → w ∈ z)}
38	35, 36, 37	3eqtr4g 2410	1 ⊢ (∀x ∈ A B ∈ C → ∩x ∈ A B = ∩{y ∣ ∃x ∈ A y = B})

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615  ∩cint 3926  ∩ciin 3970

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-int 3927  df-iin 3972

This theorem is referenced by:  dfiin2  4002