Theorem ecunres 38428

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 5942	. . 3 ⊢ ((𝑅 ∪ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
2	1	eceq2i 8664	. 2 ⊢ [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
3		ecun 38427	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴)) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))
4	2, 3	eqtrid 2778	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ∪ cun 3895   ↾ cres 5616  [cec 8620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624

This theorem is referenced by:  ecuncnvepres  38429