Theorem ssintrab 4913

Description: Subclass of the intersection of a restricted class abstraction. (Contributed by NM, 30-Jan-2015.)

Assertion

Ref	Expression
ssintrab	⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem ssintrab

Step	Hyp	Ref	Expression
1		df-rab 3390	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
2	1	inteqi 4893	. . 3 ⊢ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
3	2	sseq2i 3951	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
4		impexp 450	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ⊆ 𝑥)))
5	4	albii 1821	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ⊆ 𝑥)))
6		ssintab 4907	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ⊆ 𝑥))
7		df-ral 3052	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ⊆ 𝑥)))
8	5, 6, 7	3bitr4i 303	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥))
9	3, 8	bitri 275	1 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   ∈ wcel 2114  {cab 2714  ∀wral 3051  {crab 3389   ⊆ wss 3889  ∩ cint 4889

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-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-ss 3906  df-int 4890

This theorem is referenced by:  knatar  7312  harval2  9921  pwfseqlem3  10583  elrgspnlem4  33306  ldgenpisyslem3  34309  topjoin  36547