Theorem iinabrex 32550

Description: Rewriting an indexed intersection into an intersection of its image set. (Contributed by Thierry Arnoux, 15-Jun-2024.)

Assertion

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

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

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem iinabrex

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

Step	Hyp	Ref	Expression
1		nfra1 3266	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵
2		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑡 ∈ 𝑧
3		eleq2 2823	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑡 ∈ 𝑧 ↔ 𝑡 ∈ 𝐵))
4		vex 3463	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5	4	a1i 11	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 → 𝑧 ∈ V)
6		rspa 3231	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑡 ∈ 𝐵)
7	1, 2, 3, 5, 6	elabreximd 32491	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) → 𝑡 ∈ 𝑧)
8	7	ex 412	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 → (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧))
9	8	alrimiv 1927	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧))
10	9	adantl 481	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵) → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧))
11		nfra1 3266	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉
12	2	nfci 2886	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑧
13		nfre1 3267	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 = 𝐵
14	13	nfab 2904	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
15	12, 14	nfel 2913	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
16	15, 2	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)
17	16	nfal 2323	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)
18	11, 17	nfan 1899	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧))
19		rspa 3231	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
20	19	elexd 3483	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
21	20	adantlr 715	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
22		simplr 768	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) ∧ 𝑥 ∈ 𝐴) → ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧))
23		rspe 3232	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
24		tbtru 1548	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ⊤))
25	23, 24	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ⊤))
26	25	ex 412	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ⊤)))
27	26	alrimiv 1927	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ⊤)))
28	27	adantl 481	. . . . . . . 8 ⊢ (((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ⊤)))
29		elabgt 3651	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ ∀𝑦(𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ⊤))) → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ⊤))
30		tbtru 1548	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ⊤))
31	29, 30	sylibr 234	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ ∀𝑦(𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ⊤))) → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
32	21, 28, 31	syl2anc 584	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
33		eleq1 2822	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
34	33, 3	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → ((𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧) ↔ (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝐵)))
35	34	spcgv 3575	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧) → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝐵)))
36	35	imp 406	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝐵))
37	36	imp 406	. . . . . . 7 ⊢ (((𝐵 ∈ V ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) ∧ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) → 𝑡 ∈ 𝐵)
38	21, 22, 32, 37	syl21anc 837	. . . . . 6 ⊢ (((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) ∧ 𝑥 ∈ 𝐴) → 𝑡 ∈ 𝐵)
39	38	ex 412	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) → (𝑥 ∈ 𝐴 → 𝑡 ∈ 𝐵))
40	18, 39	ralrimi 3240	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)) → ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵)
41	10, 40	impbida 800	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)))
42	41	abbidv 2801	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)})
43		df-iin 4970	. . 3 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵}
44	43	a1i 11	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵})
45		df-int 4923	. . 3 ⊢ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)}
46	45	a1i 11	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑡 ∈ 𝑧)})
47	42, 44, 46	3eqtr4d 2780	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ⊤wtru 1541   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃wrex 3060  Vcvv 3459  ∩ cint 4922  ∩ ciin 4968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-v 3461  df-int 4923  df-iin 4970

This theorem is referenced by:  intimafv  32688