Theorem ssiin 5007

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 2918	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5006	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 208  ∀wral 3070   ⊆ wss 3899  ∩ ciin 4944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-11 2185  ax-12 2206  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ral 3071  df-v 3450  df-ss 3916  df-iin 4946

This theorem is referenced by:  triin  5218  cflim2  10210  ptbasfi  23614  limciun  25929  clsint2  36637  fnemeet2  36675  dihglblem4  41869  dihglblem6  41912  iooiinicc  46066  iooiinioc  46080  iinhoiicc  47196  smfsuplem1  47333  iinglb  49391  iineqconst2  49393