Theorem intiin 4996

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 4886	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
2		df-iin 4931	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
3	1, 2	eqtr4i 2766	1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥

Syntax hints:   = wceq 1547  {cab 2718  ∀wral 3054  ∩ cint 4884  ∩ ciin 4929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-ral 3055  df-int 4885  df-iin 4931

This theorem is referenced by:  trint  5204  relint  5769  intpreima  7018  ixpint  8870  firest  17393  efger  19691  subdrgint  20782  rintopn  22899  intcld  23030  iundifdifd  32657  iundifdif  32658  intxp  49329