Theorem ecun 39142

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 4254	. . 3 ⊢ ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)}
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)})
3		dfec2 8700	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑥 ∣ 𝐴𝑅𝑥})
4		dfec2 8700	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑆 = {𝑥 ∣ 𝐴𝑆𝑥})
5	3, 4	uneq12d 4116	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]𝑅 ∪ [𝐴]𝑆) = ({𝑥 ∣ 𝐴𝑅𝑥} ∪ {𝑥 ∣ 𝐴𝑆𝑥}))
6		elecALTV 39020	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ 𝐴(𝑅 ∪ 𝑆)𝑥))
7	6	elvd 3456	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ 𝐴(𝑅 ∪ 𝑆)𝑥))
8		brun 5156	. . . 4 ⊢ (𝐴(𝑅 ∪ 𝑆)𝑥 ↔ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥))
9	7, 8	bitrdi 290	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ∪ 𝑆) ↔ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)))
10	9	eqabdv 2893	. 2 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = {𝑥 ∣ (𝐴𝑅𝑥 ∨ 𝐴𝑆𝑥)})
11	2, 5, 10	3eqtr4rd 2806	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = ([𝐴]𝑅 ∪ [𝐴]𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {cab 2738  Vcvv 3450   ∪ cun 3897   class class class wbr 5103  [cec 8695

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-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8699

This theorem is used by:  ecunres  39143