Theorem iinxprg 5052

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 2850	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
3		iinxprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
4	3	eleq2d 2850	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐸))
5	2, 4	ralprg 4657	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)))
6	5	abbidv 2829	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶} = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)})
7		df-iin 4957	. 2 ⊢ ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶}
8		df-in 4094	. 2 ⊢ (𝐷 ∩ 𝐸) = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)}
9	6, 7, 8	3eqtr4g 2823	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080   ∩ cin 4092  {cpr 4590  ∩ ciin 4955

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-iin 4957

This theorem is used by:  pmapmeet  41021  diameetN  42304  dihmeetlem2N  42547  dihmeetcN  42550  dihmeet  42591  iinfprg  50376  infsubc2  50378