Theorem ssiin 5024

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 2931	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5023	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 209  ∀wral 3085   ⊆ wss 3913  ∩ ciin 4961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-v 3465  df-ss 3930  df-iin 4963

This theorem is referenced by:  triin  5239  cflim2  10246  ptbasfi  23706  limciun  26021  clsint2  36728  fnemeet2  36766  dihglblem4  41960  dihglblem6  42003  iooiinicc  46149  iooiinioc  46163  iinhoiicc  47279  smfsuplem1  47416  iinglb  49484  iineqconst2  49486