Theorem ssiin 4998

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 2898	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 4997	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 206  ∀wral 3051   ⊆ wss 3889  ∩ ciin 4934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-v 3431  df-ss 3906  df-iin 4936

This theorem is referenced by:  triin  5209  cflim2  10185  ptbasfi  23546  limciun  25861  clsint2  36511  fnemeet2  36549  dihglblem4  41743  dihglblem6  41786  iooiinicc  45972  iooiinioc  45986  iinhoiicc  47102  smfsuplem1  47239  iinglb  49297  iineqconst2  49299