Theorem iinrab2 5012

Description: Indexed intersection of a restricted class abstraction. (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 4951	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∩ 𝑥 ∈ ∅ {𝑦 ∈ 𝐵 ∣ 𝜑})
2		0iin 5006	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ {𝑦 ∈ 𝐵 ∣ 𝜑} = V
3	1, 2	eqtrdi 2787	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = V)
4	3	ineq1d 4159	. . . 4 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = (V ∩ 𝐵))
5		inv1 4338	. . . . 5 ⊢ (𝐵 ∩ V) = 𝐵
6	5	ineqcomi 4151	. . . 4 ⊢ (V ∩ 𝐵) = 𝐵
7	4, 6	eqtrdi 2787	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = 𝐵)
8		rzal 4434	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
9		rabid2 3422	. . . . 5 ⊢ (𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
10		ralcom 3265	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
11	9, 10	bitr2i 276	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ 𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
12	8, 11	sylib 218	. . 3 ⊢ (𝐴 = ∅ → 𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
13	7, 12	eqtrd 2771	. 2 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
14		iinrab 5011	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
15	14	ineq1d 4159	. . 3 ⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵))
16		ssrab2 4020	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ⊆ 𝐵
17		dfss 3908	. . . 4 ⊢ ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ⊆ 𝐵 ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵))
18	16, 17	mpbi 230	. . 3 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵)
19	15, 18	eqtr4di 2789	. 2 ⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
20	13, 19	pm2.61ine 3015	1 ⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}

Syntax hints:   = wceq 1542   ≠ wne 2932  ∀wral 3051  {crab 3389  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  ∩ ciin 4934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-in 3896  df-ss 3906  df-nul 4274  df-iin 4936

This theorem is referenced by: (None)