Theorem ssiin 5002

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 2894	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5001	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 206  ∀wral 3047   ⊆ wss 3897  ∩ ciin 4940

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 2113  ax-9 2121  ax-11 2160  ax-12 2180  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-v 3438  df-ss 3914  df-iin 4942

This theorem is referenced by:  triin  5212  cflim2  10154  ptbasfi  23496  limciun  25822  clsint2  36371  fnemeet2  36409  dihglblem4  41344  dihglblem6  41387  iooiinicc  45590  iooiinioc  45604  iinhoiicc  46720  smfsuplem1  46857  iinglb  48861  iineqconst2  48863