Theorem ssiin 5013

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 2924	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5012	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 208  ∀wral 3076   ⊆ wss 3904  ∩ ciin 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-v 3456  df-ss 3921  df-iin 4952

This theorem is referenced by:  triin  5224  cflim2  10220  ptbasfi  23638  limciun  25953  clsint2  36686  fnemeet2  36724  dihglblem4  41918  dihglblem6  41961  iooiinicc  46115  iooiinioc  46129  iinhoiicc  47245  smfsuplem1  47382  iinglb  49440  iineqconst2  49442