Theorem ecunres 39150

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 5991	. . 3 ⊢ ((𝑅 ∪ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
2	1	eceq2i 8743	. 2 ⊢ [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
3		ecun 39149	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴)) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))
4	2, 3	eqtrid 2809	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∪ cun 3900   ↾ cres 5661  [cec 8698

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 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8702

This theorem is used by:  ecuncnvepres  39151