Theorem intabs 5238

Description: Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)

Hypotheses

Ref	Expression
intabs.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
intabs.2	⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → (𝜑 ↔ 𝜒))
intabs.3	⊢ (∩ {𝑦 ∣ 𝜓} ⊆ 𝐴 ∧ 𝜒)

Assertion

Ref	Expression
intabs	⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑}

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜑,𝑦   𝜓,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem intabs

Step	Hyp	Ref	Expression
1		sseq1 3992	. . . . . 6 ⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → (𝑥 ⊆ 𝐴 ↔ ∩ {𝑦 ∣ 𝜓} ⊆ 𝐴))
2		intabs.2	. . . . . 6 ⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → (𝜑 ↔ 𝜒))
3	1, 2	anbi12d 632	. . . . 5 ⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → ((𝑥 ⊆ 𝐴 ∧ 𝜑) ↔ (∩ {𝑦 ∣ 𝜓} ⊆ 𝐴 ∧ 𝜒)))
4		intabs.3	. . . . 5 ⊢ (∩ {𝑦 ∣ 𝜓} ⊆ 𝐴 ∧ 𝜒)
5	3, 4	intmin3 4897	. . . 4 ⊢ (∩ {𝑦 ∣ 𝜓} ∈ V → ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓})
6		intnex 5234	. . . . 5 ⊢ (¬ ∩ {𝑦 ∣ 𝜓} ∈ V ↔ ∩ {𝑦 ∣ 𝜓} = V)
7		ssv 3991	. . . . . 6 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ V
8		sseq2 3993	. . . . . 6 ⊢ (∩ {𝑦 ∣ 𝜓} = V → (∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓} ↔ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ V))
9	7, 8	mpbiri 260	. . . . 5 ⊢ (∩ {𝑦 ∣ 𝜓} = V → ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓})
10	6, 9	sylbi 219	. . . 4 ⊢ (¬ ∩ {𝑦 ∣ 𝜓} ∈ V → ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓})
11	5, 10	pm2.61i 184	. . 3 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓}
12		intabs.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
13	12	cbvabv 2889	. . . 4 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
14	13	inteqi 4873	. . 3 ⊢ ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦 ∣ 𝜓}
15	11, 14	sseqtrri 4004	. 2 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑥 ∣ 𝜑}
16		simpr 487	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑) → 𝜑)
17	16	ss2abi 4043	. . 3 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑}
18		intss 4890	. . 3 ⊢ ({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)})
19	17, 18	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ 𝜑} ⊆ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)}
20	15, 19	eqssi 3983	1 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  Vcvv 3495   ⊆ wss 3936  ∩ cint 4869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3497  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-int 4870

This theorem is referenced by:  dfnn3  11646