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 2922	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5012	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209  ∀wral 3076   ⊆ wss 3896  ∩ ciin 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-v 3452  df-ss 3897  df-iin 4953

This theorem is used by:  triin  5228  cflim2  10348  ptbasfi  23909  limciun  26223  clsint2  37213  fnemeet2  37251  dihglblem4  42463  dihglblem6  42506  iooiinicc  46662  iooiinioc  46676  iinhoiicc  47792  smfsuplem1  47929  iinglb  50041  iineqconst2  50043