Theorem intiin 4946

Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
intiin	⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥

Distinct variable group:   𝑥,𝐴

Proof of Theorem intiin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfint2 4840	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
2		df-iin 4884	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
3	1, 2	eqtr4i 2824	1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥

Syntax hints:   = wceq 1538  {cab 2776  ∀wral 3106  ∩ cint 4838  ∩ ciin 4882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-ral 3111  df-int 4839  df-iin 4884

This theorem is referenced by:  trint  5152  relint  5656  intpreima  6815  ixpint  8472  firest  16698  efger  18836  subdrgint  19575  rintopn  21514  intcld  21645  iundifdifd  30325  iundifdif  30326