Theorem viin 5023

Description: Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2239 and abid2 2898. When 𝐴 = 𝑥, this evaluates to ∅ by intiin 5018 and intv 5326. (Contributed by NM, 11-Sep-2008.)

Assertion

Ref	Expression
viin	⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem viin

Step	Hyp	Ref	Expression
1		df-iin 4954	. 2 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴}
2		ralv 3477	. . 3 ⊢ (∀𝑥 ∈ V 𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴)
3	2	abbii 2828	. 2 ⊢ {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
4	1, 3	eqtri 2784	1 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}

This proof depends on syntax axioms:  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  Vcvv 3451  ∩ 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-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-v 3453  df-iin 4954

This theorem is used by: (None)