Theorem iinxprg 4601

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 2687	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
3		iinxprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
4	3	eleq2d 2687	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐸))
5	2, 4	ralprg 4234	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)))
6	5	abbidv 2741	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶} = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)})
7		df-iin 4523	. 2 ⊢ ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶}
8		df-in 3581	. 2 ⊢ (𝐷 ∩ 𝐸) = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)}
9	6, 7, 8	3eqtr4g 2681	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912   ∩ cin 3573  {cpr 4179  ∩ ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436  df-un 3579  df-in 3581  df-sn 4178  df-pr 4180  df-iin 4523

This theorem is referenced by:  pmapmeet  35059  diameetN  36345  dihmeetlem2N  36588  dihmeetcN  36591  dihmeet  36632