Theorem ssiin 5009

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 2896	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5008	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 206  ∀wral 3049   ⊆ wss 3899  ∩ ciin 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-v 3440  df-ss 3916  df-iin 4947

This theorem is referenced by:  triin  5219  cflim2  10171  ptbasfi  23523  limciun  25849  clsint2  36472  fnemeet2  36510  dihglblem4  41496  dihglblem6  41539  iooiinicc  45730  iooiinioc  45744  iinhoiicc  46860  smfsuplem1  46997  iinglb  49009  iineqconst2  49011