Theorem ssiin 5004

Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
ssiin	⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem ssiin

Step	Hyp	Ref	Expression
1		nfcv 2891	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5003	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 206  ∀wral 3044   ⊆ wss 3903  ∩ ciin 4942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-v 3438  df-ss 3920  df-iin 4944

This theorem is referenced by:  triin  5215  cflim2  10157  ptbasfi  23466  limciun  25793  clsint2  36303  fnemeet2  36341  dihglblem4  41276  dihglblem6  41319  iooiinicc  45523  iooiinioc  45537  iinhoiicc  46655  smfsuplem1  46792  iinglb  48806  iineqconst2  48808