Theorem iunxprg 5096

Description: A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)

Hypotheses

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

Assertion

Ref	Expression
iunxprg	⊢ ((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 iunxprg

Step	Hyp	Ref	Expression
1		df-pr 4629	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		iuneq1 5008	. . . 4 ⊢ ({𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶)
3	1, 2	ax-mp 5	. . 3 ⊢ ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶
4		iunxun 5094	. . 3 ⊢ ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶)
5	3, 4	eqtri 2765	. 2 ⊢ ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶)
6		iunxprg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
7	6	iunxsng 5090	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷)
8	7	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷)
9		iunxprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
10	9	iunxsng 5090	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸)
11	10	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸)
12	8, 11	uneq12d 4169	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) = (𝐷 ∪ 𝐸))
13	5, 12	eqtrid 2789	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∪ cun 3949  {csn 4626  {cpr 4628  ∪ ciun 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-iun 4993

This theorem is referenced by:  iunxunpr  32580  ovnsubadd2lem  46660  rnfdmpr  47293