Theorem iinrab2 4519

Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Assertion

Ref	Expression
iinrab2	⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iinrab2

Step	Hyp	Ref	Expression
1		iineq1 4471	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∩ 𝑥 ∈ ∅ {𝑦 ∈ 𝐵 ∣ 𝜑})
2		0iin 4514	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ {𝑦 ∈ 𝐵 ∣ 𝜑} = V
3	1, 2	syl6eq 2660	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = V)
4	3	ineq1d 3775	. . . 4 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = (V ∩ 𝐵))
5		incom 3767	. . . . 5 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
6		inv1 3922	. . . . 5 ⊢ (𝐵 ∩ V) = 𝐵
7	5, 6	eqtri 2632	. . . 4 ⊢ (V ∩ 𝐵) = 𝐵
8	4, 7	syl6eq 2660	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = 𝐵)
9		rzal 4025	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
10		rabid2 3096	. . . . 5 ⊢ (𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
11		ralcom 3079	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
12	10, 11	bitr2i 264	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ 𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
13	9, 12	sylib 207	. . 3 ⊢ (𝐴 = ∅ → 𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
14	8, 13	eqtrd 2644	. 2 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
15		iinrab 4518	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
16	15	ineq1d 3775	. . 3 ⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵))
17		ssrab2 3650	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ⊆ 𝐵
18		dfss 3555	. . . 4 ⊢ ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ⊆ 𝐵 ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵))
19	17, 18	mpbi 219	. . 3 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵)
20	16, 19	syl6eqr 2662	. 2 ⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
21	14, 20	pm2.61ine 2865	1 ⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}

Syntax hints:   = wceq 1475   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∩ ciin 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-iin 4458

This theorem is referenced by: (None)