Theorem iunxprg 5056

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 4587	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		iuneq1 4968	. . . 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 5054	. . 3 ⊢ ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶)
5	3, 4	eqtri 2783	. 2 ⊢ ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶)
6		iunxprg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
7	6	iunxsng 5050	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷)
8	7	adantr 486	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷)
9		iunxprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
10	9	iunxsng 5050	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸)
11	10	adantl 487	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸)
12	8, 11	uneq12d 4116	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) = (𝐷 ∪ 𝐸))
13	5, 12	eqtrid 2807	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∪ cun 3897  {csn 4584  {cpr 4586  ∪ ciun 4951

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587  df-iun 4953

This theorem is used by:  iunxunpr  33043  ovnsubadd2lem  47476  rnfdmpr  48172