Theorem ecunres 38761

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

Assertion

Ref	Expression
ecunres	⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))

Proof of Theorem ecunres

Step	Hyp	Ref	Expression
1		resundir 5946	. . 3 ⊢ ((𝑅 ∪ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
2	1	eceq2i 8676	. 2 ⊢ [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
3		ecun 38760	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴)) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))
4	2, 3	eqtrid 2786	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ∪ cun 3881   ↾ cres 5620  [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:  ecuncnvepres  38762