Theorem iinxprg 4045

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 2301	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
3		iinxprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
4	3	eleq2d 2301	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐸))
5	2, 4	ralprg 3720	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)))
6	5	abbidv 2349	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶} = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)})
7		df-iin 3973	. 2 ⊢ ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦 ∈ 𝐶}
8		df-in 3206	. 2 ⊢ (𝐷 ∩ 𝐸) = {𝑦 ∣ (𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸)}
9	6, 7, 8	3eqtr4g 2289	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510   ∩ cin 3199  {cpr 3670  ∩ ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-sn 3675  df-pr 3676  df-iin 3973

This theorem is referenced by: (None)