Theorem ssintrab 4937

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 3417	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
2	1	inteqi 4917	. . 3 ⊢ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
3	2	sseq2i 3967	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
4		impexp 455	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ⊆ 𝑥)))
5	4	albii 1849	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ⊆ 𝑥)))
6		ssintab 4931	. . 3 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ⊆ 𝑥))
7		df-ral 3080	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ⊆ 𝑥)))
8	5, 6, 7	3bitr4i 306	. 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥))
9	3, 8	bitri 278	1 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   ∈ wcel 2143  {cab 2741  ∀wral 3079  {crab 3416   ⊆ wss 3906  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-ss 3923  df-int 4914

This theorem is referenced by:  knatar  7357  harval2  9984  pwfseqlem3  10646  elrgspnlem4  33546  ldgenpisyslem3  34536  topjoin  36857