Theorem ecun 38644

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 4262	. . 3 ⊢ ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)}
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)})
3		dfec2 8648	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑥 ∣ 𝐴𝑅𝑥})
4		dfec2 8648	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑆 = {𝑥 ∣ 𝐴𝑆𝑥})
5	3, 4	uneq12d 4123	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]𝑅 ∪ [𝐴]𝑆) = ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}))
6		elecALTV 38522	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ 𝐴(𝑅 ∪ 𝑆)𝑥))
7	6	elvd 3448	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ 𝐴(𝑅 ∪ 𝑆)𝑥))
8		brun 5151	. . . 4 ⊢ (𝐴(𝑅 ∪ 𝑆)𝑥 ↔ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥))
9	7, 8	bitrdi 287	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)))
10	9	eqabdv 2870	. 2 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)})
11	2, 5, 10	3eqtr4rd 2783	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = ([𝐴]𝑅 ∪ [𝐴]𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114  {cab 2715  Vcvv 3442   ∪ cun 3901   class class class wbr 5100  [cec 8643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647

This theorem is referenced by:  ecunres  38645