Theorem iinrab 4954

Description: Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)

Assertion

Ref	Expression
iinrab	⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})

Distinct variable groups:   𝑦,𝐴,𝑥   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem iinrab

Step	Hyp	Ref	Expression
1		r19.28zv 4404	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜑)))
2	1	abbidv 2862	. 2 ⊢ (𝐴 ≠ ∅ → {𝑦 ∣ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜑)})
3		df-rab 3115	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)}
4	3	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)})
5	4	iineq2i 4903	. . 3 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)}
6		iinab 4953	. . 3 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)}
7	5, 6	eqtri 2821	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)}
8		df-rab 3115	. 2 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜑)}
9	2, 7, 8	3eqtr4g 2858	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  {crab 3110  ∅c0 4243  ∩ ciin 4882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-nul 4244  df-iin 4884

This theorem is referenced by:  iinrab2  4955  riinrab  4969  ubthlem1  28653  pmapglbx  37065  preimageiingt  43355  preimaleiinlt  43356