Theorem iunxprg 3946

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 3583	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		iuneq1 3879	. . . 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 3945	. . 3 ⊢ ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶)
5	3, 4	eqtri 2186	. 2 ⊢ ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶)
6		iunxprg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
7	6	iunxsng 3941	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷)
8	7	adantr 274	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷)
9		iunxprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
10	9	iunxsng 3941	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸)
11	10	adantl 275	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸)
12	8, 11	uneq12d 3277	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) = (𝐷 ∪ 𝐸))
13	5, 12	syl5eq 2211	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136   ∪ cun 3114  {csn 3576  {cpr 3577  ∪ ciun 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-iun 3868

This theorem is referenced by:  unct  12375