Theorem intiin 5017

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 4908	. 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
2		df-iin 4953	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
3	1, 2	eqtr4i 2786	1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥

This proof depends on syntax axioms:   = wceq 1569  {cab 2738  ∀wral 3076  ∩ cint 4906  ∩ ciin 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-ral 3077  df-int 4907  df-iin 4953

This theorem is used by:  trint  5229  relint  5796  intpreima  7071  ixpint  8957  firest  17609  efger  19938  subdrgint  21066  rintopn  23233  intcld  23364  iundifdifd  33204  iundifdif  33205  intxpd  50037