Theorem iinrab2 5028

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 4969	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∩ 𝑥 ∈ ∅ {𝑦 ∈ 𝐵 ∣ 𝜑})
2		0iin 5022	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ {𝑦 ∈ 𝐵 ∣ 𝜑} = V
3	1, 2	eqtrdi 2812	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = V)
4	3	ineq1d 4165	. . . 4 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = (V ∩ 𝐵))
5		inv1 4348	. . . . 5 ⊢ (𝐵 ∩ V) = 𝐵
6	5	ineqcomi 4157	. . . 4 ⊢ (V ∩ 𝐵) = 𝐵
7	4, 6	eqtrdi 2812	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = 𝐵)
8		rzal 4450	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
9		rabid2 3445	. . . . 5 ⊢ (𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
10		ralcom 3291	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
11	9, 10	bitr2i 279	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ 𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
12	8, 11	sylib 221	. . 3 ⊢ (𝐴 = ∅ → 𝐵 = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
13	7, 12	eqtrd 2796	. 2 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
14		iinrab 5027	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
15	14	ineq1d 4165	. . 3 ⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵))
16		ssrab2 4028	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ⊆ 𝐵
17		dfss 3918	. . . 4 ⊢ ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ⊆ 𝐵 ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵))
18	16, 17	mpbi 233	. . 3 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} = ({𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} ∩ 𝐵)
19	15, 18	eqtr4di 2814	. 2 ⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
20	13, 19	pm2.61ine 3039	1 ⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2956  ∀wral 3077  {crab 3413  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ∩ ciin 4952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4280  df-iin 4954

This theorem is used by: (None)