Theorem ssiin 5019

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 2926	. 2 ⊢ Ⅎ𝑥𝐶
2	1	ssiinf 5018	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209  ∀wral 3080   ⊆ wss 3900  ∩ ciin 4957

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-v 3456  df-ss 3901  df-iin 4959

This theorem is used by:  triin  5235  cflim2  10380  ptbasfi  23955  limciun  26269  clsint2  37332  fnemeet2  37370  dihglblem4  42620  dihglblem6  42663  iooiinicc  46827  iooiinioc  46841  iinhoiicc  47957  smfsuplem1  48094  iinglb  50264  iineqconst2  50266