Theorem ecun 38578

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 4260	. . 3 ⊢ ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)}
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)})
3		dfec2 8638	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑥 ∣ 𝐴𝑅𝑥})
4		dfec2 8638	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑆 = {𝑥 ∣ 𝐴𝑆𝑥})
5	3, 4	uneq12d 4121	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]𝑅 ∪ [𝐴]𝑆) = ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}))
6		elecALTV 38464	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ 𝐴(𝑅 ∪ 𝑆)𝑥))
7	6	elvd 3446	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ 𝐴(𝑅 ∪ 𝑆)𝑥))
8		brun 5149	. . . 4 ⊢ (𝐴(𝑅 ∪ 𝑆)𝑥 ↔ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥))
9	7, 8	bitrdi 287	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)))
10	9	eqabdv 2869	. 2 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)})
11	2, 5, 10	3eqtr4rd 2782	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = ([𝐴]𝑅 ∪ [𝐴]𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2113  {cab 2714  Vcvv 3440   ∪ cun 3899   class class class wbr 5098  [cec 8633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637

This theorem is referenced by:  ecunres  38579