Theorem intiin 5023

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 4912	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
2		df-iin 4958	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
3	1, 2	eqtr4i 2755	1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥

Syntax hints:   = wceq 1540  {cab 2707  ∀wral 3044  ∩ cint 4910  ∩ ciin 4956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-ral 3045  df-int 4911  df-iin 4958

This theorem is referenced by:  trint  5232  relint  5782  intpreima  7042  ixpint  8898  firest  17395  efger  19648  subdrgint  20712  rintopn  22796  intcld  22927  iundifdifd  32490  iundifdif  32491  intxp  48820