Theorem iinxprg 5092

Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)

Hypotheses

Ref	Expression
iinxprg.1	⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
iinxprg.2	⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)

Assertion

Ref	Expression
iinxprg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸

Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iinxprg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iinxprg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
2	1	eleq2d 2818	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
3		iinxprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
4	3	eleq2d 2818	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐸))
5	2, 4	ralprg 4698	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)))
6	5	abbidv 2800	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶} = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)})
7		df-iin 5000	. 2 ⊢ ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶}
8		df-in 3955	. 2 ⊢ (𝐷 ∩ 𝐸) = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)}
9	6, 7, 8	3eqtr4g 2796	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  ∀wral 3060   ∩ cin 3947  {cpr 4630  ∩ ciin 4998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-un 3953  df-in 3955  df-sn 4629  df-pr 4631  df-iin 5000

This theorem is referenced by:  pmapmeet  38948  diameetN  40231  dihmeetlem2N  40474  dihmeetcN  40477  dihmeet  40518