Theorem intabs 5261

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 3942	. . . . . 6 ⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → (𝑥 ⊆ 𝐴 ↔ ∩ {𝑦 ∣ 𝜓} ⊆ 𝐴))
2		intabs.2	. . . . . 6 ⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → (𝜑 ↔ 𝜒))
3	1, 2	anbi12d 630	. . . . 5 ⊢ (𝑥 = ∩ {𝑦 ∣ 𝜓} → ((𝑥 ⊆ 𝐴 ∧ 𝜑) ↔ (∩ {𝑦 ∣ 𝜓} ⊆ 𝐴 ∧ 𝜒)))
4		intabs.3	. . . . 5 ⊢ (∩ {𝑦 ∣ 𝜓} ⊆ 𝐴 ∧ 𝜒)
5	3, 4	intmin3 4904	. . . 4 ⊢ (∩ {𝑦 ∣ 𝜓} ∈ V → ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓})
6		intnex 5257	. . . . 5 ⊢ (¬ ∩ {𝑦 ∣ 𝜓} ∈ V ↔ ∩ {𝑦 ∣ 𝜓} = V)
7		ssv 3941	. . . . . 6 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ V
8		sseq2 3943	. . . . . 6 ⊢ (∩ {𝑦 ∣ 𝜓} = V → (∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓} ↔ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ V))
9	7, 8	mpbiri 257	. . . . 5 ⊢ (∩ {𝑦 ∣ 𝜓} = V → ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓})
10	6, 9	sylbi 216	. . . 4 ⊢ (¬ ∩ {𝑦 ∣ 𝜓} ∈ V → ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓})
11	5, 10	pm2.61i 182	. . 3 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑦 ∣ 𝜓}
12		intabs.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
13	12	cbvabv 2812	. . . 4 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
14	13	inteqi 4880	. . 3 ⊢ ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦 ∣ 𝜓}
15	11, 14	sseqtrri 3954	. 2 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ ∩ {𝑥 ∣ 𝜑}
16		simpr 484	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑) → 𝜑)
17	16	ss2abi 3996	. . 3 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑}
18		intss 4897	. . 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 3933	1 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  Vcvv 3422   ⊆ wss 3883  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-int 4877

This theorem is referenced by:  dfnn3  11917