Theorem ecun 38760

Description: The union coset of 𝐴. (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
ecun	⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = ([𝐴]𝑅 ∪ [𝐴]𝑆))

Proof of Theorem ecun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unab 4236	. . 3 ⊢ ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)}
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)})
3		dfec2 8636	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑥 ∣ 𝐴𝑅𝑥})
4		dfec2 8636	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑆 = {𝑥 ∣ 𝐴𝑆𝑥})
5	3, 4	uneq12d 4099	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]𝑅 ∪ [𝐴]𝑆) = ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}))
6		elecALTV 38638	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ 𝐴(𝑅 ∪ 𝑆)𝑥))
7	6	elvd 3437	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ 𝐴(𝑅 ∪ 𝑆)𝑥))
8		brun 5123	. . . 4 ⊢ (𝐴(𝑅 ∪ 𝑆)𝑥 ↔ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥))
9	7, 8	bitrdi 288	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)))
10	9	eqabdv 2872	. 2 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)})
11	2, 5, 10	3eqtr4rd 2785	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = ([𝐴]𝑅 ∪ [𝐴]𝑆))

Syntax hints:   → wi 4   ↔ wb 207   ∨ wo 853   = wceq 1547   ∈ wcel 2119  {cab 2717  Vcvv 3431   ∪ cun 3881   class class class wbr 5072  [cec 8631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635

This theorem is referenced by:  ecunres  38761