Theorem intiin 5018

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 4909	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
2		df-iin 4954	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
3	1, 2	eqtr4i 2787	1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥

This proof depends on syntax axioms:   = wceq 1570  {cab 2739  ∀wral 3077  ∩ cint 4907  ∩ ciin 4952

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 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-ral 3078  df-int 4908  df-iin 4954

This theorem is used by:  trint  5230  relint  5797  intpreima  7068  ixpint  8946  firest  17596  efger  19925  subdrgint  21053  rintopn  23220  intcld  23351  iundifdifd  33149  iundifdif  33150  intxpd  49911