Theorem ecunres 39043

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 5993	. . 3 ⊢ ((𝑅 ∪ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
2	1	eceq2i 8733	. 2 ⊢ [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
3		ecun 39042	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴)) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))
4	2, 3	eqtrid 2810	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143   ∪ cun 3903   ↾ cres 5663  [cec 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692

This theorem is referenced by:  ecuncnvepres  39044