Theorem ssiin 5014

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 2923	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5013	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209  ∀wral 3077   ⊆ wss 3899  ∩ ciin 4952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-v 3453  df-ss 3916  df-iin 4954

This theorem is used by:  triin  5229  cflim2  10334  ptbasfi  23893  limciun  26207  clsint2  37097  fnemeet2  37135  dihglblem4  42334  dihglblem6  42377  iooiinicc  46523  iooiinioc  46537  iinhoiicc  47653  smfsuplem1  47790  iinglb  49901  iineqconst2  49903