Theorem dfiin2g 4991

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 3080	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑤 ∈ 𝐵))
2		df-ral 3080	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
3		clel4g 3625	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐶 → (𝑤 ∈ 𝐵 ↔ ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧)))
4	3	imim2i 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) → (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐵 ↔ ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧))))
5	4	pm5.74d 276	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧))))
6	5	alimi 1834	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) → ∀𝑥((𝑥 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧))))
7		albi 1841	. . . . . . 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))))
8	6, 7	syl 18	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧))))
9	2, 8	sylbi 220	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧))))
10		df-ral 3080	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑤 ∈ 𝑧) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ∈ 𝑧)))
11	10	albii 1842	. . . . . . 7 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑤 ∈ 𝑧) ↔ ∀𝑧∀𝑥(𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ∈ 𝑧)))
12		alcom 2196	. . . . . . 7 ⊢ (∀𝑥∀𝑧(𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ∈ 𝑧)) ↔ ∀𝑧∀𝑥(𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ∈ 𝑧)))
13	11, 12	bitr4i 281	. . . . . 6 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑤 ∈ 𝑧) ↔ ∀𝑥∀𝑧(𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ∈ 𝑧)))
14		r19.23v 3192	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑤 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑤 ∈ 𝑧))
15		vex 3461	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
16		eqeq1 2769	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵))
17	16	rexbidv 3189	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
18	15, 17	elab 3641	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
19	18	imbi1i 352	. . . . . . . 8 ⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑤 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑤 ∈ 𝑧))
20	14, 19	bitr4i 281	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑤 ∈ 𝑧) ↔ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑤 ∈ 𝑧))
21	20	albii 1842	. . . . . 6 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑤 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑤 ∈ 𝑧))
22		19.21v 1962	. . . . . . 7 ⊢ (∀𝑧(𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ∈ 𝑧)) ↔ (𝑥 ∈ 𝐴 → ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧)))
23	22	albii 1842	. . . . . 6 ⊢ (∀𝑥∀𝑧(𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ∈ 𝑧)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧)))
24	13, 21, 23	3bitr3ri 305	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑧(𝑧 = 𝐵 → 𝑤 ∈ 𝑧)) ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑤 ∈ 𝑧))
25	9, 24	bitrdi 290	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → 𝑤 ∈ 𝐵) ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑤 ∈ 𝑧)))
26	1, 25	bitrid 286	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑤 ∈ 𝑧)))
27	26	abbidv 2831	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑤 ∣ ∀𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} = {𝑤 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑤 ∈ 𝑧)})
28		df-iin 4955	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑤 ∣ ∀𝑥 ∈ 𝐴 𝑤 ∈ 𝐵}
29		df-int 4909	. 2 ⊢ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑤 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑤 ∈ 𝑧)}
30	27, 28, 29	3eqtr4g 2825	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   = wceq 1563   ∈ wcel 2145  {cab 2743  ∀wral 3079  ∃wrex 3089  ∩ cint 4908  ∩ ciin 4953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-v 3459  df-int 4909  df-iin 4955

This theorem is referenced by:  dfiin2  4993  iinexg  5309  dfiin3g  5950  iinfi  9365  mreiincl  17638  iinopn  23020  clsval2  23168  alexsublem  24162